UC-NRLF 


GIFT  OF 
Spreckles 


/'' 


Digitized  by  the  Internet  Archive 
t  ^ .  in  2007  with  funding  from 
;  IViicrosoft  Corporation 


httpymww:waiim^ 


Stu&ie0  in  ISueineea 


First  Series,  No.  3 


.THE  ACCOUNTANCY  OF  INVESTMENT. 

INCI.UDING  A  TREATISE  ON  COMPOUND 
INTEREST,  ANNUITIES,  AMORTISATION, 
AND  THE    VAI.UATION  OF    SECURITIES 


BY  CHARLES  EZRA  SPRAGUE,  A.M.,  PH.D.,  C.P.A.,  PROFESSOR 
IN  THE  NEW  YORI^  UNIVERSITY  SCHOOL  OF  COMMERCE, 
ACCOUNTS  AND  FINANCE ;  PRESIDENT  OF  THE  UNION  DIME 
SAVINGS  INSTITUTION  ;  CHAIRMAN  OF  THE  SAVINGS  BANK 
SECTION  OF  THE  AMERICAN  BANKERS  ASSOCIATION     :     :     : 


Published  fok.  the  New  York  University  School  of  Commerce, 

Accounts   and    Finance  by  the  Business   Publishino  Company 

New  York,  1904 


^ 


v^f^l 


SPREGKELS 


Copyright,  1904,  by  Charles  E.  Spragub 


TRUNK    BROS. 

85  WILL-AM   STREET 

NEW  YORK 


PREFACE. 

The  following  chapters  embrace  the  substance  of  lectures  delivered 
before  the  classes  of  the  New  York  University  Schooi<  oe  Commerce, 
Accounts  and  Finance.  They  have  been  in  many  places  condensed, 
and  in  others  expanded,  with  a  view  to  their  use  as  a  text-book. 

I  have  introduced  a  treatise  on  Interest,  Discount,  Annuities,  Sink- 
ing Funds,  Amortisation  and  Valuation  of  Bonds,  as  I  had  not  been  able 
to  find  any  suitable  text-book  which  I  could  recommend.  I  hope  that 
this  will  be  useful  to  many  who  desire  to  inaugurate  more  scientific 
methods  in  their  accountancy,  but  are  unable  to  find  intelligible  rules  for 
the  computations.  Treatises  on  the  subject  written  for  actuarial  students 
are  invariably  too  difficult,  except  for  those  who  have  not  only  been  highly 
trained  in  algebra,  but  are  fresh  in  its  use,  and  this  makes  the  subject 
forbidding  to  many  minds.  I  have  made  all  my  demonstrations  arith- 
metical and  illustrative,  but,  I  think,  none  the  less  convincing  and  intel- 
ligible. 

I  am  indebted  to  Prof.  Joseph  Hardcastle,  C.P.A.,  and  to  Walter 
B.  Hallett,  A.B.,  for  valuable  suggestions  and  assistance. 

CHARLES  E.  SPRAGUE. 
New  York,  November,  1904. 


TABLE  OF  CONTENTS. 


INTRODUCTORY  CHAPTER. 
Thk  Theory  of  Accounts. 

PAGB 

Balance  Sheet 5 

Equation  of  Accountancy 5 

Examples 6 

Ledger 8 

Transactions 10 

Debit  and  Credit 11 

CHAPTER  I. 
Capitai,  and  Revenue. 

Definition  of  Capital 12 

Use  of  Capital 12 

Sources  of  Capital 12 

Investment 13 

Revenue 13 

Interest ;  Rent ;  Dividends 13 

CHAPTER  II. 

Interest. 

Time 15 

Ratio  of  Increase 17 

Amount 17 

Present  Worth 18 

Abbreviated  Rules 18 

CHAPTER  III. 

The  Use  op  I^ogarithms. 

Nature  of  Logarithms 20 

Rules  for  their  use 20 

Example  of  use  in  Compound  Interest 21 

CHAPTER  IV. 
Amount  of  an  Annuity. 

Definition  and  Example  of  Annuity 24 

Abbreviated  Summation '. .  25 

Rule 26 

CHAPTER  V. 
Present  Worth  of  an  Annuity. 

Abbreviated  Process 27 

Example  and  Demonstration 28 

Schedule  of  Annuity 28 

Number  of  Years  Purchase 29 


CHAPTER  VI. 
Rent  of  Annuity  and  Sinking  Fund. 

PAGE 

Process  of  Finding  Rent  when  Present  Worth  is  Given 30 

Sinking  Fund 31 

Relation  between  Sinking  Fund  and  Rent  of  Annuity 31 


CHAPTER  VII. 
NoMiNAi.  AND  Effective  Rates. 

Periods  not  Always  Annual 32 

Rate  per  Annum,  in  that  Case,  only  Nominal 32 

Process  of  Reducing  Nominal  to  Effective  Rates 32 

Process  of  Reducing  Effective  to  Nominal  Rates 33 


CHAPTER  VIII. 

Valuation  of  Bonds. 

Definitions 34 

Customary  Features 34 

Designation 34 

Premium  and  Discount 35 

Cash  Rate  and  Income  Rate 35 

Evaluation  of  Bond  :  First  Method 35 

Second  Method 36 

Schedule  of  Amortisation,  A 37 

Schedule  of  Accumulation,  B 39 

Use  of  Tables 39 

Schedules  C  and  D 40 

Purchases  between  Interest  Dates 41 

Schedule  E 42 

Balancing  Periods  not  Coinciding  with  Interest  Periods 42 

Schedule  F 43 

Eliminating  Residues  :  First  Method 44 

Second  Method 45 

Third  Method 46 

Schedule  H 46 

Short  Terminals 46 

Schedule  1 47 

Discounting 47 

Serial  Bonds 48 

Trust  Funds 49 

Justice  Cullen 50 

Single  Column  Schedule 52 

Irredeemable  Bonds 53 

Optional  Redemption 53 


CHAPTER  IX. 
Forms  of  Account — Generai,  Principles. 

General  and  Subordinate  Ledgers 55 

Accounting  for  Interest  as  Fast  as  Earned 65 


CHAPTER  X. 
Reai,  Estate  Mortgages.  page 

Definitions 57 

Requisites  for  Accountancy 57 

Form  of  Ledger 68 

Register  of  Interest  Due 60 

Models  of  Mortgage  Ledger 61 — 64 

Mortgages  Account,  General  Ledger 66 

Classified  Registers  of  Mortgages 67 

CHAPTER  XI. 
lyOANS  ON  Collateral. 

Directions 68 

Model  Account 69 

CHAPTER  XII. 

Interest  Accounts. 

Stages  in  Interest 70 

Daily  Register  of  Interest  Accruing 71 

Monthly  Summary  of  Interest  Accruing; 72 

General  Ledger  Accounts  ;  Interest  Revenue,   Interest  Accrued  and 

Interest  Due 72—73 

CHAPTER  XIII. 
Bonds  and  Similar  Securities. 

Book  Value  Only  ;  Three-Column  Ledger 74 

Bond  Sales 75 

Exhibiting  Par  and  Cost ;  Scalar  Ledger 76 

Model  of  Scalar  Account 77 

Amortisation  Entries 78 

General  Ledger  ;  Two  Plans  for  Keeping  Par,  Cost  and  Book  Value . .  79 

Irredeemable  Bonds 82 

Models  for  General  Ledger  Account :  Plan  1 83 

Plann,! 84 

Plan  II,  2 85 

Plan  II,  3 86 

Plan  II,  4  (Balance  Column  Form)  87 

CHAPTER  XIV. 
Discounted  Values. 

Rate  of  Interest  and  Rate  of  Discount 88 

Discount  as  an  Offset 89 

Progressive  Example 90 

Adjustment  at  Balancing  Periods 90 

APPENDIX  I. 

Logarithms,  to  12  Places,  of  Various  Ratios 91 

APPENDIX  II. 

Multipliers  for  Finding  Prices  of  Quarterly  Bonds 92 

(See  note,  p.  54.) 

APPENDIX  III. 

Summary  of  Compound  Interest  Processes 93 


^.,x 


ie^. 


Wk 


THE  ACCOUNTANCY  OF  INVESTMENT. 


INTRODUCTORY  CHAPTER. 

Theory  of  Accounts. 

The  Balance  Sheet  of  a  business  expresses  the  status 
of  that  business  at  a  certain  point  of  time.  It  normally  contains 
three  classes  of  values:  assets,  liabilities  and  proprietorship, 
and  expresses  an  equation  between  one  of  these  classes,  the 
assets,  and  the  two  others.  As  a  fact,  stripped  of  all  tech- 
nicality, the  assets  are  always  exactly  equal  to  the  sum  of  the 
liabilities  and  the  proprietorship : 

Assets  =  Liabilities  +  Proprietorship. 
This  is  merely  an  inverted  way  of  defining  proprietorship  as 
being  the  excess  of  assets  over  liabilities  : 

Assets  —  Liabilities  =  Proprietorship. 
In  practice  we  generally  write  the  equation  in  the  form  of  a 
ledger  account : 


Assets  : 


[Equal  Totals] 


LiABiiviTiES  : 


Proprietorship 


As  the  proprietorship  is  simply  the  excess  of  assets  over 
liabilities,  it  is  evident  that  at  any  moment,  if  the  fads  are  all 
ascertained,  the  same  equation  must  hold  good ;  it  must  be  per- 
petually true  that,  through  all  shiftings  and  changes  : 

Assets  =  Liabilities  -|-  Proprietorship. 
This  is  the  equation  of  accountancy,  and  all  the  processes  of 
bookkeeping  depend  upon  it. 


The  Accountancy  of  Investment. 


The  following  is  a  condensed  balance  sheet  of  the  affairs 
of  an  individual  in  business: 

Bai^anc^  Sheet  of  Wii.i,iam  Smith. 


Cash 

15,082.34 

Bills  Payable 

18,000.00 

Merchandise 

17,082.65 

Personal  Creditors 

5,465.35 

Personal  Debtors 

8,123.17 

Bills  Receivable 

7,000.00 

William  Smith 

23,822.81 

$37,288.16 

137,288.16 

The  assets  are  partly  composed  of  property,  actually  in 
possession,  and  partly  of  debts  due  to  Smith;  while  the 
liabilities  are  entirely  indebtedness  due  by  Smith. 

The  net  proprietorship  is  designated  by  the  name  of  the 
proprietor,  although  this  is  not  universally  the  case. 

Instead  of  a  single  proprietor  there  may  be  a  partnership, 
and  the  proprietorship  may  be  represented  either  separately 
or  in  aggregate,  as  follows : 

Bai^ance  Sheet  oe  Jones  &  Smith. 


Cash 

$8,589.09 

Bills  Payable 

18,000.00 

Merchandise 

39,249.38 

Personal  Creditors 

5,465.35 

Bills  Receivable 

7,000.00 

Mortgage  Payable 

4,000.00 

Personal  Debtors 

24,095.32 

James  Jones 

47,643.63 

Real  Estate 

10.000.00 

William  Smith 

23,822  81 

188,933.79 

188,933.79 

BAI.ANCE  Sheet  o 

E  Jones  &  Smith. 

Cash 

18,589.09 

Bills  Payable 

$8,000.00 

Merchandise 

39,249.38 

Personal  Creditors 

5,465.35 

Bills  Receivable 

7,000.00 

Mortgage  Payable 

4,000.00 

Personal  Debtors 
Real  Estate 

24,095.32 
10,000.00 

Jones  &  Smith 

{their  joint  capital) 

71,468.44 

188,933.79 

$88,933.79 

Theory  of  Accounts. 


I^et  it  be  supposed  that  Messrs.  Tones  &  Smith,  instead 
of  a  partnership,  had  preferred  to  form  a  company,  named  the 
Jones  Mercantile  Company.  They  consider  that,  as  the  actual 
value  of  their  joint  proprietorship  is  over  $71,000,  it  would  be 
quite  proper  to  capitalize  it  at  $60,000,  in  600  shares,  of  $100 
each.  Nevertheless,  there  is  a  total  proprietorship  of 
$71,468.44,  as  before,  all  of  which  must  be  represented  in  some 
form. 

In  order  to  represent  both  the  amount  of  the  capitalization 
and  the  true  proprietary  value,  we  divide  the  total  proprietor- 
ship into  two  parts :  — 

Capital :  par,  or  face  value  of  shares  |60,000.00 

Surplus  :  excess  of  real  value  over  par  11,468.44 

Their  sum  is  the  real  proprietorship      |71,468.44 

The  resulting  balance  sheet  would  be  :  — 

BAI.ANCK  Sheet  oe  the  Jones  Mercantii^e  Company. 


Cash 

$8,589.09 

Bills  Payable 

18,000.00 

Merchandise 

39,249.38 

Personal  Creditors 

5,465.35 

Bills  Receivable 

7,000.00 

Mortgage  Payable 

4,000.00 

Personal  Debtors 

24,095.32 

j  Capital  Stock 
\  Surplus 

60,000.00 

Real  Estate 

10,000.00 
$88,933.79 

11,468.44 

188,933.79 

These  are  the  usual  forms  of  the  balance  sheet.  As  our 
debtors  are  all  on  the  left  hand  side,  and  our  creditors  are  on  the 
right,  it  has  become  customary  to  call  the  left  the  dedttside,  and 
the  right,  the  credit  side,  notwithstanding  the  debit  side  con- 
tains much  more  than  debtors  and  the  creditor  side  much  more 
than  creditors. 

A  better  term  for  the  debit  side  of  the  balance  sheet  is  the 
Active  ;  and  for  the  credit  side,  the  Passive. 

The  Passive  has  two  widely  different  sets  of  values  :  the 
liabilities  and  the  proprietorship,  and  I  see  no  advantage  in 
stretching  the  term  ' '  liabilities ' '  to  cover  both.  To  call  the 
proprietorship  a  liability  is  purely  a  technicality ;  the  part 
owned  is  precisely  that  part  which  is/ree  from  liability. 


8 


Th:^  Accountancy  of  Investment. 


The  balance  sheet  presents  the  status  at  some  certain  point 
of  time.  We  need  also  some  means  for  recording  what  occurs, 
for  the  changes  which  take  place  between  the  balance  sheets. 
For  this  purpose,  the  ledger  is  opened,  being  a  system  of 
accounts  ;  one  account  at  least  for  each  line  of  the  balance  sheet. 
It  was  formerly  supposed  that  these  accounts  must  be  kept  in 
one  invariable  form,  regardless  of  their  nature,  such  form  being 
substantially  that  of  the  balance  sheets  shown  above. 
Although  this  is  no  longer  the  rule  we  will  employ  the  tra- 
ditional form  in  this  illustration. 

A  miniature  ledger  made  up  from  the  balance  sheet  of  the 
Jones  Mercantile  Company  would  begin  somewhat  after  this 

fashion: 

Cash. 


Balance 


18,589.09 


BiivivS  Payabi^e. 


Balance 


18,000.00 


Mkrchandisk. 


Balance 


139,249.38 


BiiviyS  Receivable. 


Balance 


17,000.00 


Theory  of  Accounts. 


"Personal  Debtors"  would  probably  comprise  a  number 
of  accounts,  headed  by  the  name  of  each  debtor. 

Cr, 


Dr. 


A.  B. 


Balance 


' '  Personal  Creditors ' '  would  likewise  comprise  a  number 
of  separate  accounts. 

Dr.  M.  N.  Cr. 


Balance 


REAi,  Estate. 


Balance 


110,000.00 


Dr. 


Mortgage  Payabi^e. 


Cr. 


Balance 


14,000.00 


CapitaIv  Stock. 


Balance 


160,000.00 


lo  Th^  Accountancy  of  Investment. 

SURPI^US. 


Balance  |11,468.44 


Thus  the  balance  sheet  has  been  dissected  and  accounts 
have  been  opened  for  each  department  of  the  business.  A 
single  account  might  have  been  opened  for  '  *  Personal  Debtors, ' ' 
but  in  that  case  it  would  have  to  be  expanded  into  subordinate 
accounts,  so  as  to  give  information  as  to  each  debtor.  The  same 
is  true  of  "  Personal  Creditors,"  which  is  also  a  group  account. 

For  good  reasons  the  balance  with  which  each  account 
begins  is  placed  on  the  same  side  as  it  occupied  in  the  balance 
sheet.  Any  increase  of  that  balance  will  also  be  entered  on  the 
same  side,  and  any  decrease  on  the  other  side. 

Any  possible  transaction  will  increase  the  debit  (left  hand) 
side  of  the  ledger,  and  its  credit  (right  hand)  side  to  exactly 
the  same  amount.     Let  us  consider  some  of  the  possible  cases : 

1.  If  any  asset  is  increased,  we  must  either 

part  with  some  other  asset, 
or  run  into  debt, 
or,  (if  neither  of  these  is  true, ) 
our  wealth  is  increased. 

That  is,  an  increase  of  assets  is  attended  by  a  decrease  of  assets, 
or  an  increase  of  liability,  or  an  increase  of  proprietorship,  one 
or  more  of  them. 

2.  If  a  liability  is  increased,  we  must  either 

receive  some  asset, 

or  pay  off  some  other  liability, 

or  else  we  have  lost. 
Therefore,  an  increase  of  liability  is  attended  by  an  increase  of 
assets,  by  a  decrease  of  liability,  or  a  decrease  of  proprietorship. 
As  there  are  three  elements  in  the  accounts  —  assets, 
liabilities  and  proprietorship  —  and  as  each  of  these  may  be  in- 
creased or  decreased,  there  are  six  possible  entries,  at  least  two 
of  which  arise  from  every  transaction,  as  follows  : 


Thkory  of  Accounts.  ii 

Debits.  Credits. 

Increase  of  Assets.  Decrease  of  Assets. 

Decrease  of  lyiabilities.  Increase  of  lyiabilities. 

Decrease  of  Proprietorship.  Increase  of  Proprietorship. 


The  increase  and  decrease  of  proprietorship  is  called 
Profit  and  Loss.  As  it  is  of  the  utmost  importance  to  study 
the  causes  of  Profit  and  Loss,  certain  subsidiary  accounts  are 
opened  for  the  sole  purpose  of  classifying  profits  and  losses 
according  to  their  sources.  At  the  time  of  constructing  the  next 
balance  sheet,  these  subsidiary  accounts  (Interest,  Rent, 
Expenses,  Sales,  etc. )  are  transferred  to  a  general  Profit  and 
Loss  account,  which  presents  an  analysis  of  the  conduct  of  the 
business  for  the  period  elapsed.  This  Profit  and  Loss  account 
is  in  turn  transferred  to  the  permanent  Proprietary  Accounts. 


The  foregoing  is  a  sketch  of  the  general  theory  of  Double 
Entry  Bookkeeping.  There  are  various  ways  of  looking  at  the 
subject,  but  I  think  this  the  most  direct  and  best  suited  to  the 
present  purpose.  For  some  other  purposes,  such  as  what  may 
be  called  juridical  accounts,  the  respective  rights  and  obligations 
of  the  parties  are  the  basis,  rather  than  the  struggle  to  increase 
proprietorship.     The  equation  for  this  purpose  would  be  : 

Charge  =  Discharge  +  Accountability. 

In  general  the  equation  comprises  not  two,  but  three,  terms : 

Positives  =  Negatives  -}-  Resultant. 


12  Th^  Accountancy  of  Invkstmbnt. 


CHAPTER  I. 

Capitai.  and  Rkvsnus. 

1. — Capital  is  that  portion  of  wealth  which  is  set  aside  for 
the  production  of  additional  wealth.  The  capital  of  a  business, 
therefore,  is  the  whole  or  a  part  of  the  assets  of  the  business, 
and,  of  course,  appears  on  the  active  or  debit  side  of  its  balance 
sheet.  This  is  the  sense  in  which  the  word  '  *  capital ' '  is  used 
in  economics;  but  in  bookkeeping  the  words  "capital  account" 
are  often  used  in  quite  another  sense  to  mean  accounts  on 
the  credit  or  passive  side  which  denote  proprietorship.  To 
prevent  confusion,  I  will  avoid  the  use  of  this  expression, 
capital  account. 

2.— Use  of  Capital.  In  active  business  capital  must  be 
employed  —  must  be  combined  with  skill  and  industry  to  pro- 
duce more  wealth.  Businesses,  and  consequently  their  ac- 
counting methods,  vary  as  to  the  manner  in  which  capital  is 
used.  Cash  is  potential  capital  of  all  kinds,  as  desired.  In  a 
manufacturing  business  it  is  exchanged  for  machinery,  appli- 
ances, raw  materials,  labor,  which  transforms  the  material  into 
the  product.  In  mercantile  business  it  is  expended  for  goods, 
bought  at  one  price,  to  sell  at  another,  and  for  collecting,  dis- 
playing, caring  for,  advertising  and  delivering  the  goods.  To 
bridge  over  the  time  between  selling  and  collecting,  additional 
capital  is  required,  usually  known  as  "working  capital,"  but 
which  might  be  more  appropriately  styled  ' '  waiting  capital. ' ' 
Thus  we  may  analyse  each  kind  of  business  and  show  that  its 
capital  assets  depend  upon  the  character  of  the  business. 

3. — Sources  of  Capital.  On  the  other  side  of  the  balance 
sheet  the  capital  must  be  accounted  for  as  to  who  furnishes  it. 
Here  there  are  two  sharply  divided  classes  :  loan-capital  or 
liability,  and  own  capital  or  proprietorship.  The  great  dis- 
tinction is  that  the  latter  participates  in  the  profits  and  bears 
the  losses,  while  the  former  takes  its  share  irrespective  of  the 
success  of  the  concern.  It  is  the  own-capital  which  is  referred 
to  in  the  phrase  "  capital  account." 


Capital  and  Revenue.  13 

•4. — While  we  often  speak  of  a  man's  capital  as  being 
invested  in  the  business,  yet  when  we  use  the  word  more  strict- 
ly, we  confine  it  to  the  non-participating  sense.  Thus  we  say, 
he  not  only  owns  the  business,  but  he  has  some  investments 
besides.  In  the  strictest  sense,  then,  investment  implies  divest- 
ing one's  self  of  the  possession  and  control  of  one's  assets  and 
granting  such  possession  and  control  to  another.  The  advan- 
tage of  the  use  of  the  capital  must  be  great  enough  to  enable 
the  user  to  earn  more  than  the  sum  which  he  pays  to  the  in- 
vestor or  capitalist.  There  are  many  cases  where  the  surrender 
is  not  absolute,  and  there  is  more  or  less  risk  assumed  by  the 
investor.  This  I  should  call  not  absolute  investment,  but  to 
some  extent  partnership.  The  essence  of  strict  investment  is 
vicarious  earning,  a  share  of  the  gain  not  dependent  on  the 
fortunes  of  the  handler. 

5. — Revenue.  All  investment  is  made  with  a  view  to 
revenue,  which  is  the  share  of  the  earnings  given  for  the  use 
of  capital.  This  takes  three  forms:  interest,  rent  and  divi- 
dends —  the  former  two  corresponding  to  strict  investment,  and 
the  latter  to  participation. 

6. — Interest  and  rent  do  not  essentially  differ.  Both  are 
stipulated  payments  for  the  use  of  capital ;  but  in  the  latter 
the  same  physical  asset  must  be  returned  on  the  completion 
of  the  contract.  If  you  borrow  a  dollar,  you  may  repay  any 
dollar  you  please ;  if  you  hire  a  hous6  or  a  horse,  you  may  not 
return  any  house  or  any  horse,  but  must  produce  the  identical 
one  you  had.   Interest  and  rent  are  both  proportionate  to  time. 

7. — Dividends.  These  are  profits  paid  over  to  the  owners 
of  the  own-capital,  whether  partners  or  shareholders.  The 
amount  is  subtracted  from  the  collective  assets  and  paid  over 
to  the  separate  owners.  Theoretically  there  is  no  profit  nor 
loss  in  this  distribution.  I  have  more  cash,  but  my  share 
in  the  collective  assets  is  exactly  that  much  less.  It  is  dis- 
tributed, partly  because  it  is  needed  by  the  participants  for 
consumption ;  partly  because  no  more  capital  can  be  profitably 
used  in  the  enterprise.  Some  concerns,  for  example  some 
banks,  which  can  profitably  use  more  capital,  and  whose  share- 
holders do  not  require  it  for  consumption,  refrain  from  dividing, 


14  I^HS  Accountancy  of  Invkstmejnt. 

and  the  accumulation  inures  just  as  surely  to  the  shareholders, 
and  is  realizable  through  increased  value  of  the  shares  upon 
sale.  Thus,  dividends  are  not  strictly  revenue.  Yet  the  share- 
holder may  treat  them  as  such ;  the  dividend  may  be  so 
regular  as  practically  to  be  fixed,  or  his  shares  may  be  prefer- 
ential, so  that  to  some  extent  he  is  receiving  an  ascertained 
amount  ;  or,  as  in  case  of  a  leased  railway,  it  may  be  fixed  by 
contract.  Still,  legally  speaking,  the  dividend  is  instantaneous, 
and  does  not  accrue,  like  interest  and  rent. 

8. — As  all  investment  is  really  the  buying  of  revenue,  and 
as  the  value  of  the  investment  depends  largely  upon  the  amount 
of  revenue,  and  as  the  typical  form  of  revenue  is  interest,  it  is, 
therefore,  necessary  to  study  the  laws  of  interest,  including 
those  more  complex  forms — annuities,  sinking  funds  and  amor- 
tisation. Although  there  is  a  special  branch  of  accountancy  — 
the  actuarial — which  deals  not  only  with  these  subjects,  but 
with  life  and  other  contingencies,  yet  it  is  very  necessary  for 
the  general  accountant  to  understand  at  least  the  principles  of 
the  subject. 


The  Accountancy  op  Investment.  15 

.     •  CHAPTER  II. 

Interest. 
9. — The  elements  of  interest  are  rate,  time  and  principal. 

10. — The  time  is  divided  into  periods  ;  at  the  end  of  each 
period  a  certain  sum  for  each  unit  of  the  principal  is  payable. 
The  ratio  between  the  unit  of  principal  and  the  sum  paid  for' 
its  use  is  the  rate,  and  is  expressed  in  hundredths  or  "per 
centum. ' '  Thus,  if  the  contract  is  to  pay  three  cents  for  each 
dollar  of  principal  each  year,  it  may  be  expressed,  .03  per 
annum,  3  per  centum  per  annum,  3  per  cent.,  or  3%.  Where 
the  period  is  not  yearly,  but  a  less  time,  it  is  customary  to 
speak,  nevertheless,  of  the  annual  rate.  Thus,  instead  of  3% 
per  half  year,  we  say  6%,  payable  semi-annually.  Instead  of 
i%  per  quarter,  we  say  4%,  quarterly.  In  our  discussions  of 
interest,  however,  we  shall  treat  of  periods  and  of  the  rate  per 
period,  in  order  to  avoid  complication. 

11. — As  the  law  does  not  recognize  interest  for  any  fraction 
of  a  day,  it  becomes  necessary  to  inquire  what  is  meant  by  a 
half  year  or  a  quarter.  The  Statutory  Construction  Law 
(Chapter  677,  Laws  of  1892,  §25)  solves  this  difficulty  by  pre- 
scribing that  a  half  year  is  not  1S2}4  days,  but  six  calendar 
months,  and  that  a  quarter  is  not  91^  days,  but  three  calendar 
months. 

Calendar  months  are  computed  as  follows  :  —  Commence 
at  the  day  from  which  the  reckoning  is  made,  excluding  that 
day  ;  then  the  day  in  the  next  month  having  the  same  number 
will  at  its  close  complete  the  first  month  ;  the  second  month 
will  end  with  the  same  numbered  day,  and  so  on  to  the  same 
day  of  the  final  month.  One  difficulty  arises  :  Suppose  we 
have  started  with  the  31st  and  the  last  month  has  only  30  days 
or  less.  Then,  the  law  says,  the  month  ends  with  the  last  day. 
One  month  from  January  31st,  1904,  was  February  29th  ;  one 
month  from  January  30th  or  January  29th  would  also  terminate 
on  February  29th ;  in  a  common  year,  not  a  leap  year,  the  last 
day  would  be  February  28th. 

12. — A  Day  also  requires  definition.  The  legal  day  begins 
and  ends  at  midnight.  In  reckoning  from  one  day  to  another 
you  must  not  include  the  day  from  which.     Thus,  if  a  loan 


1 6  The  Accountancy  of  Investment. 

is  made  at  any  hour  on  the  third  of  the  month  and  paid  at 
any  hour  on  the  fourth,  there  is  one  day's  interest,  and  that 
one  day  is  the  fourth,  not  the  third.  Practically  it  is  the 
nights  that  count.  If  five  mid-nights  have  passed  since  the 
loan  was  made,  then  there  is  five  days'  interest  accrued. 

13. — Parts  of  a  period.  In  practice  any  fraction  of  an 
interest  period  is  computed  at  the  corresponding  fraction  of 
the  rate,  although  theoretically  this  is  not  quite  just.  If 
the  regular  period  is  a  year,  then  the  odd  days  should  be 
reckoned  as  365ths  of  a  year.  Also,  if  the  contract  is  for  days 
only  and  there  is  no  mention  of  months,  quarters  or  half  years, 
then  also  a  day  is  regarded  as  ^J^  of  a  year.  But  when  the 
contract  is  for  months,  quarters  or  half  years,  the  fractional 
time  must  be  divided  into  months.  Finally  we  have  the  odd 
days  left  over,  and  doubt  exists  as  to  how  they  should  be 
treated. 

Before  1892  there  was  no  doubt.  The  statute  distinctly 
stated  that  a  number  of  days  less  than  a  month  should  be 
estimated  for  interest  as  30ths  of  a  month,  or,  consequently, 
360ths  of  a  year.  This  was  a  most  excellent  provision  and 
merely  enacted  what  had  been  the  custom  long  before.  The 
so-called  "  360  day  "  interest  tables  are  based  upon  this  rule. 
But  the  revisers  of  the  statutes  of  the  State,  in  1892,  dropped 
this  sensible  provision  and  left  the  question  open.  No  judicial 
decision  has  since  been  rendered  on  the  subject,  but  many  good 
lawyers  think  that  the  odd  days  must  be  computed  as  365 ths  of 
a  year.  In  business  nearly  every  one  calls  the  odd  days  360ths, 
and  it  is  only  in  legal  accountings  that  there  can  be  any  question. 
It  would  be  well  if  the  old  provision  could  be  re-enacted  or 
re-established  by  the  courts.  If  it  is  necessary  to  correct  the 
interest  on  the  odd  days  from  360ths  to  365ths,  it  may  readily 
be  done  by  subtracting  from  such  interest  if^  of  itself. 

14. — Interest  is  assumed  to  be  paid  when  due.  If  it  is 
not  so  paid,  it  ought  to  be  added  to  the  principal  and  interest 
be  computed  on  the  increased  principal.  But  as  the  law  does 
not  directly  sanction  this,  simple  interest  is  spoken  of  as  if  it 
were  a  distinct  species,  where  the  original  principal  remains 
unchanged,  even  though  interest  is  in  default.  There  really  is 
no  such  thing,  for  the  interest  money  which  is  wrongfully  with- 


Interest.  17 

held  by  the  borrower  may  be  by  him  employed,  and  thus  com- 
pound interest  is  earned  ;  only  the  wrong  man  gets  it.  All  the 
calculations  of  finance  depend  upon  compound  i7iterest,  which 
is  the  only  rational  and  consistent  method.  When  I  have 
occasion  to  speak  of  the  interest  for  one  period  I  shall  call  it 
"  single  interest." 

15. — If  we  add  to  1  the  decimal  denoting  the  rate,  we  have 
the  ratio  of  increase.  Thus,  if  the  rate  is  .03  per  period,  1.03 
is  the  ratio  of  increase,  or  simply  the  ratio,  or  the  multiplier. 

16.— The  Amount  is  the  principal  and  interest  taken 
together.  At  the  end  of  the  first  period  the  amount  of  $1 .  00 
at  3%  interest  is  $1 .  03.  Instead  of  considering  the  $1 .  00  and 
the  3  cents  as  two  separate  sums  to  be  added  together,  it  is  best 
to  consider  the  operation  as  the  single  one  of  multiplying  $1 .  00 
by  the  ratio  1 .  03. 

17. — The  principal  which  is  employed  during  the  second 
period  is  $1 .  03.  It  is  evident  that  this,  like  the  original  $1 .  00, 
should  be  multiplied  by  the  ratio  1.03.  The  new  amount  will 
be  the  square  of  1 .  03,  which  we  may  write    1 .  03  x  1 .  03 

or,    (1.03)2 

or,     1.0609 
This  is  the  new  amount  on  interest  during  the  third  period. 
At  the  end  of  the  third  period  the  amount  will  be 

1.03x1.03x1.03 

or,     1.033 
1.092727 
At  the  end  of  the  fourth  period  we  have  reached  the  amount, 

1.03* 

or,  1.12550881 
We  here  find  that  the  number  of  decimals  is  becoming  unwieldy, 
and  conclude  to  cut  it  down.  If  we  desire  to  limit  it  to  seven 
figures  we  reject  the  1,  rounding  the  result  off  to  1.1255088; 
if  we  prefer  to  use  only  six  figures,  we  round  it  «/  to  1 .  125509, 
which  is  nearer  than  1 .  125508. 

18. — Thus  the  amount  of  $1 .00  at  the  end  of  any  number 
of  periods  is  obtained  by  taking  such  a  power  of  the  ratio  of 
increase  as  is  indicated  by  the  number  of  periods  ;  or  by  mul- 
tiplying $1 .  00  by  the  ratio  as  many  times  as  the  number  of 
periods.     The  remainder,  after  subtracting  the  original  prin- 


1 8       The  Accountancy  of  Investment. 

cipal,  is  the  compound  interest.    Thus  the  compound  interest 
for  four  periods  is  .125509.     The  single  interest  is  .03. 

19. — The  present  worth  of  a  future  sum  is  a  smaller  sum, 
which,  put  at  interest,  will  amount  to  the  future  sum.  The 
present  worth  of  $1 .  00  is  such  a  sum  that  $1 .  00  will  be  its 
amount.  Using  the  same  suppositions  as  before  we  desire  to 
find  such  a  number  as,  when  multiplied  by  1.03,  will  amount 
to  $1 .  00  in  four  periods.  $1 .  00  must,  therefore,  be  divided  by 
1 .  03  for  the  first  period. 

1.03  )  1.00000000  (   .970873 
927 

730 
721 

900 

824 

760 
721 

390 
309 

81 

The  result,  rounded  up  at  the  6th  place,  is  .  970874,  the 
present  worth  of  $1  at  3%  for  one  period,  or  j^,  or  1  -^  1.03. 
The  present  worth  for  two  periods  may  be  obtained  either  by 
again  dividing  .970874  by  1.03,  or  by  multiplying  .970874 
by  itself,  or  by  dividing  1  by  1.0609,  each  of  which  operations 
gives  the  same  result,  .  942596  =  ^2  The  third  term  is 
—33  =  .915142,  and  the  fourth  is  ^^.  =  .888487. 

20. — If  we  arrange  these  four  results  in  reverse  order  fol- 
lowed by  $1  and  by  the  amounts  computed  in  article  17,  we  have 
a  continuous  series : 

.888487 

.915142 

.942596 

.970874 
1.00 
1.03 
1.0609 
1.092727 
1.125509 


Intbrest.  19 

21. — It  may  be  observed  that  each  of  these  numbers  is  an 
amount  of  every  preceding  number  and  2,  present  worth  of  every 
succeeding  number,  and  that  when  one  number  is  the  amount 
of  another,  the  latter  is  the  present  worth  of  the  former ;  in 
other  words,  that  amount  and  present  worth  are  reciprocals. 

22. — Each  one  of  these  numbers  may  be  obtained  from  the 
preceding  one  by  multiplying  by  1 .  03.  Hence,  as  multipli- 
cation is  easier  than  division,  if  we  can  obtain^8487  directly, 
we  may  supply  the  intermediate  values  more  readily.  This 
brief  process  for  finding. ^8487  will  be  explained  in  the  next 
chapter. 

23. — In  the  present  worth,  .97087  of  a  single  period,  it  is 
evident  that  the  original  |1 .  00  has  been  diminished  by  .  02913, 
which  is  a  little  less  than  .03  ;  in  fact  it  is  .03  -=- 1..03.  This 
difference  .02913  is  called  the  discount.  In  the  present  worth 
for  two  periods  the  discount  is  1 — .942596,  or  .057404.  This 
and  succeeding  discounts  for  greater  numbers  of  periods  are 
compound  discounts. 

24. — Compound  discount  does  not  bear  any  such  direct 
relation  to  compound  interest  as  single  discount  does  to  single 
interest.  It  can  only  be  found  by  first  ascertaining  the  present 
worth  and  then  subtracting  that  from  1. 

25. — We  can  reduce  the  rules  to  more  compact  form  by 
the  use  of  symbols.  Let  .y  represent  the  amount  of  1 ;  />  the 
present  worth ;  i  the  rate  of  interest  per  period ;  n  the  number 
of  periods,  and  d  the  rate  of  discount.  Let  the  compound 
interest  be  represented  by  I,  and  the  compound  discount  by  D. 

26. — Then,  by  article  15,  the  ratio  of  increase  is  (l-j-z.) 
By  article  18,  .y  =  (1  +  i)  «;  and  I  =  .y  —  1.  By  article  19, 
/  =  1  -^  (1  +  iY  ;  and  D  =  1  — /. 

27. — The  method  of  ascertaining  the  values  of  s  and  / 
through  successive  multiplications  and  divisions  is,  for  a  large 
number  of  periods,  intolerably  slow.  A  much  briefer  way, 
by  the  use  of  certain  auxiliary  numbers,  called  logarithms, 
will  be  explained  in  the  next  chapter. 


20  Thb  Accountancy  of  Investment. 


CHAPTER  III. 

The  Use  of  IvOgarithms. 

28. — For  multiplying  or  dividing  a  great  many  times  by 
the  same  number,  there  is  no  device  hitherto  invented  which  is 
superior  to  a  table  of  logarithms. 

29. — The  use  of  logarithms  does  not  require  a  knowledge 
of  the  higher  mathematics.  It  is  purely  an  arithmetical  help. 
The  popular  prejudice  to  the  effect  that  there  is  something 
mysterious  or  occult  about  logarithms  has  no  foundation. 

30. — The  ordinary  books  of  logarithms  are  calculated  to  7 
places  of  decimals,  sometimes  extended  for  certain  numbers  to 
8.  If  you  wished  to  multiply  by  1 .  03  fifty  times,  the  logarithm 
would  give  you  the  first  seven  figures  only  of  the  answer,  but  as 
the  remaining  figures  are  so  very  insignificant,  the  result  will 
for  most  questions  be  near  enough  even  if  rounded  off  at  the  6th 
place. 

31. — All  the  books  of  logarithmic  tables  contain,  in  an  in- 
troduction, rules  for  using  the  tables,  and  these  should  be  studied, 
and  the  examples  worked  out.  These  books  have  the  ordinary 
numbers  on  the  left  in  a  regular  series,  in  four  figures  only  ; 
the  fifth  figure  is  at  the  head  of  one  of  the  ten  columns  to  the 
right.  The  sixth  and  seventh  figures  are  obtained  by  a  little 
side-table. 

32. — Briefly,  the  rules  of  logarithms  are  as  follows  : 

By  adding  logarithms  you  multiply  numbers. 

By  subtracting  logarithms  you  divide  numbers. 

By  multiplying  logarithms  you  raise  numbers  to  powers. 

By  dividing  logarithms  you  extract  roots  of  numbers. 

33. — The  last  two  of  these  rules  are  the  only  ones  necessary 
to  be  employed  in  the  calculations  of  compound  interest. 

34. — In  the  common  system  of  logarithms  10  is  the  base  ; 
that  is,  the  logarithm  of  10  is  1.     The  logarithm  of  100  (being 


The  Use  of  Logarithms.  21 

two  tens  multiplied  together)  is  2.  The  logarithm  of  1,000  (in 
which  10  is  three  times  a  factor)  is  3.  We  may  thus  go  on  in- 
definitely, saying  in  abbreviated  language,  log.  10,000  =  4  ; 
log.  1,000,000  =  6.  In  all  these  cases,  the  logarithm  is  the 
number  of  zeroes  used  to  express  the  number.  What  is  the 
meaning  of  these  zeroes  ?  Each  of  them  means  that  ten,  the 
base  of  numeration,  enters  once  as  a  factor. 

1,000,000  :  1  followed  by  6  zeroes  means 

that  1  is  multiplied  6  times  by  10  ; 

or  it  may  be  written  1  x  (10)  \    Log.    1,000,000  =  6 

Similarly Log.       100,000  =  5  • 

To  multiply  these  numbers  together 

you  really  add  the  logarithms,  and 

write  1  followed  by 11 

zeroes.     Thus   there   is   a  kind   of 

logarithmic    method     in     ordinary 

arithmetic. 

35.  To  demonstrate  the  use  of  logarithms  in  compound 
interest,  let  us  take  an  example  and  work  it  out,  illustrating  each 
Step.  We  will  take  the  same  rate  as  before,  .03,  but  endeavor 
to  find  the  amount  for  50  periods. 

36. — The  ratio  of  increase  is  1.03.  We  look  for  the  log- 
arithm of  this  ratio.  At  the  top  of  page  192  (Chambers'  or 
Babbage's  tables)  we  find  this  line: 


No. 

0    12 

3 

4 

5 

6   7    8 

9 

10300 

0128  3722  4144  4566 

4987 

5409 

5831 

6252  6674  7096 

7517 

37. — The  meaning  of  this  is  that  the  logarithms  are  as 

follows  : 

log.  1.03  .  .01283722 

'•     1.03001  .01284144 

"     1.03002  .01284566 

•♦    1.03003  .01284987 


"     1.03009  .01287517 

The  first  figures  are  given  once  only,  which  saves  space  and 
time  in  searching. 

38. — Since  1.03  is  to  be  taken  as  a  factor  50  times,    we 
must  multiply  its  logarithm  by  50.     This  gives : 

50  X  .01283722=  .6418610. 


22  The  Accountancy  of  Investment. 

This  result  is  the  logarithm  of  the  answer,  for  when  we  have 
found  the  corresponding  number  we  shall  know  the  value  of 

39. — We  must  now  look  in  the  right  hand  columns  for  the 
logarithm  figures  .6418610.  We  first  look  for  the  641,  which 
stands  out  by  itself,  overhanging  a  blank  space.  This  we  find 
on  page  73,  and  w^e  find  that  the  nearest 

approach  is .  6418606 

which  is   the  logarithm  of  4.3839. 
We     now,     from     our   logarithm,  .6418610 

subtract    the  above   approximation  .6418606 

and  have  a  remainder 04 

In  the  margin  is  a  little  difference-table,  reading  thus  : 

99 


1 

10 

2 

20 

3 

30 

4 

40 

5 

50 

6 

59 

7 

69 

8 

79 

9 

89 

The  left  hand  column  represents  the  6th  figure  of  the  answe  . 
If  the  remainder  were  10,  instead  of  4,  the  next  figure  would 
be  1 ;  if  it  were  69,  the  next  figure  would  be  7.  But  is  less 
than  10,  therefore  the  6th  figure  is  0.  The  7th  figure  is  4, 
because  40  would  give  4  for  the  6th  figure. 

40.— Thus  we  have  obtained  our  result.  $4.383904  is 
the  amount  of  $1 .00  compounded  for  50  periods  at  3%.  This 
result  is  slightly  inaccurate  in  the  last  figure,  for  the  reason 
that  two  places  were  lost  by  multiplying.  Had  we  taken  the 
10  figure  logarithm  on  page  xviii  of  Chambers' ,     .  0128372247 

this  multiplied  by  50  w^ould  give 641861235 

or  rounded  off  at  the  7th  place 6418612 

From  this  subtract 6418606 

and  we  have  the  remainder 06 

which  gives  the  more  accurate  result 4.383906 

41. — As  it  is  necessary,  for  problems  involving  many  periods, 


The  Use  of  I^ogarithms.  23 

to  use  a  very  extended  logarithm,  I  give  in  Appendix  1  a  table 
of  twelve-place  logarithms  for  a  number  of  different  ratios 
of  increase  (1  -{-  i).  These  are  at  much  closer  intervals  than 
any  table  previously  published,  and,  with  a  10  figure  book  of 
logarithms,  will  give  exact  results  to  the  nearest  cent  on 
$1,000,000. 

42. — We  will  further  exemplify  the  advantage  of  the 
logarithmic  method  by  solving  a,  present- worth  problem. 
Taking  50  periods  at  3%  for  $1 .  00,  we  discount  it  as  follows  : 
Multiply  the  logarithm  of  1.03  by  50,  just  as  in  Article  40, 
giving  .641861235.  But  it  is  the  reciprocal  of  1.03^^  or 
1  -f- 1.03^°,  which  we  wish  to  obtain;  hence  we  must  subtract 
.641861235  from  the  logarithm  of  1,  which  is  0. 

0.000000000 

0.641861235 

Remainder  1.358138765 

In  subtracting  a  greater  from  a  less  logarithm,  we  get  a 
negative  whole  number  (as  shown  by  the  minus  above),  the 
decimal  part  being  positive,  and  obtained  by  ordinary 
subtraction. 

43. — Neglecting  the  1,  we  search  in  the  right  hand  column 
for  .358138765.  On  page  31  we  find  that  .3581253  is  the 
logarithm  of  2.2810. 

From 3581388 

Subtract 3581253 

and  we  have  a  remainder 135 

From  the  marginal  table  we  find  that 133 

corresponds  to  7,  hence  the  6th  figure 

is  7,  giving  so  far  the  result  2 .  28107.  

There  is  still  a  remainder  of 2 

which  by  the  table  is  equivalent  to  1  for  the  7th  figure.  Hence, 
we  have  the  full  result  .  2281071,  the  decimal  point  being  moved 
one  place  to  the  left,  as  directed  by  the  1. 


24  Tun  Accountancy  of  Invkstment. 


CHAPTER  IV. 

Amount  of  an  Annuity. 

44. — We  have  now  investigated  the  two  fundamental 
problems  in  compound  interest  :  viz. ,  to  find  the  amount  of  a 
present  worth,  and  to  find  the  present  worth  of  an  amount. 
The  next  question  is  a  more  complex  one :  to  find  the  amount 
and  the  present  worth  of  a  series  of  payments.  If  these  pay- 
ments are  irregular  as  to  time,  amount  and  rate  of  interest, 
the  only  way  is  to  make  as  many  separate  computations  as 
there  are  sums  and  then  add  them  together.  But  if  the  sums, 
times  and  rate  are  uniform,  we  can  devise  a  method  for  finding 
the  amount  or  present  worth  at  one  operation. 

Annuity.  A  series  of  payments  of  like  amount,  made 
at  regular  periods,  is  called  an  annuity,  even  though  the  period 
be  not  a  year,  but  a  half  year,  a  quarter  or  any  other  length  of 
time.  Thus,  if  an  agreement  is  made  for  the  following 
payments : 

On  Sept.  9  1904  $100. 

On  March  9  1905  100. 

On  Sept.  9  1905  100. 

and  on  March  9  1906  100 . 

this  would  be  an  annuity  of  $200  per  annum,  payable  semi- 
annually; in  other  words,  an  annuity  of  $100  per  period,  term- 
inating after  4  periods.  It  is  required  to  find  on  March  9,  1904, 
assuming  the  rate  of  interest  as  6%  per  annum,  payable  semi- 
annually {3%  per  period)  :  First,  what  will  be  the  total 
amount  to  which  the  annuity  will  have  accumulated  on  March 
9,  1906;  second,  what  is  now,  on  March  9,  1904,  the  present 
worth  of  this  series  of  future  sums.  It  is  evident  that  the 
answer  to  the  first  question  will  be  greater  than  $400,  and  that 
the  answer  to  the  second  question  will  be  less  than  $400. 

45. — It  is  easy,  in  this  case,  to  find  the  separate  amounts 
of  the  payments,  for  the  number  of  terms  is  very  small,  and 
we  have  already  computed  the  corresponding  values  of  $1 .  00. 


Amount  of  an  x\nnuity.  25 

The  last  $100  will  have  no  accumulation,  and  will  be 

merely $100. 

The  third  $100  will  have  earned  in  one  period,  $3.00, 

and  will  amount  to 103 . 

The  second  $100  will  amount  to  106.09 

The  first  $100  (rounded  off  at  cents)  will  amount  to  109.27 

and  the  total  amount  will  be $418.36 

46. — If,  however,  there  were  50  terms  instead  of  4,  the 
work  of  computing  these  50  separate  amounts,  even  by  the  use 
of  logarithms,  would  be  very  tedious. 

47. — lyet  us  write  down  the  successive  amounts  of  $1 .  00 
under  one  another: 

a 
Amounts  of  $1. 
1.00 
1.03 
1.0609 
1.092727 

48. — Now,  as  we  have  the  right  to  take  any  principal  we 
choose  and  multiply  it  by  the  number  indicating  the  value  of 
$1.00,  let  us  assume  one  dollar  and  three  cents,  and  multiply 
each  of  the  above  figures  by  1 .  03,  setting  the  products  in  a 
second  column : 

a.  b.  c. 

Amounts  of  $1.00         Amounts  of  $1.03         Amounts  of  $0.03 
1.00  1.03 

1.03  1.0609 

1.0609  -  1.092727  ■ 

1.092727  ^  1.12550881 

49. — Our  object  in  doing  this  was  by  subtracting  column 
a  from  b  to  find  the  amount  of  an  annuity  of  three  cents.  Be- 
fore subtracting,  we  have  the  right  to  throw  out  any  numbers 
which  are  identical  in  the  two  columns.  Expunging  these 
like  quantities,  we  have  left  only  the  following : 

a.  b.  c. 

Annuity  of  $1.00  Annuity  of  $1 . 03  Annuity  of  $0.03 

1.00  1.12550881 

less        1.00 


1 .  12550881  Amount  0 .  12550881 

That  is,  an  annuity  of  three  cents  will  amount,  under  the 
conditions  assumed,  to  twelve  cents  and  the  decimal  550881. 
Therefore,  an  annuity  of  07ie  cent  will  amount  to  one-third  of 


26  The  Accountancy  op  Investment. 

.12550881  or  .04183627.  An  annuity  of  $1.00  will  amount  to 
100  times  as  much,  or  $4.183627,  which  agrees  exactly  with 
the  result  obtained  by  addition,  in  Article  45. 

50.— The  number  .12550881  (obtained  by  subtracting 
1.00  from  1.12550881)  is  actually  the  compound  interest  for 
the  given  rate  and  time,  and  the  number  .03  is  the  single 
interest;  the  amount  of  the  annuity  of  $1.00  is  .12550881-^  .03 
=  4 .  183627.  This  suggests  another  way  of  looking  at  it.  The 
compound  interest  up  to  any  time  is  really  the  amount  of  a 
smaller  annuity,  one  of  three  cents  instead  of  a  dollar,  con- 
structed on  exactly  the  same  plan,  and  used  as  a  model. 

51. — Rule.  To  find  the  amount  of  an  annuity  of  $1.00 
for  a  given  time  and  rate,  divide  the  compound  interest  by  a 
single  interest,  both  expressed  decimally. 

52. — Let  S  and  P  represent  the  amount  and  the  present 
worth,  not  of  a  single  $1.00,  but  of  an  annuity  of  $1.00,  then 
S  =  I  -f-  /. 

53. — To  illustrate,  let  us  take  the  case  worked  out  in  Article 
40,  where  we  found  the  amount  of  a  single  dollar  at  3%,  for  50 

periods  to  be 4.383906 

Subtracting  one  dollar 1.000000 

The  compound  interest  is .     3.383906 

Divide  this  by  .03  and  we  have .' 112.79687 

which  is  the  amount  to  which  50  payments  of  $1.00  each,  at 
3%  per  period,  would  accumulate. 


Thk  Accountancy  of  Investment 


27 


CHAPTER  V. 

Present  Worth  of  an  Annuity. 

54. — To  find  the  present  worth  of  an  annuity,  we  can,  of 
course,  find  the  present  worth  of  each  payment  and  add  them 
together  ;  but  it  will  evidently  save  a  great  deal  of  labor  if  we 
can  derive  the  present  worth  immediately,  as  we  have  learned 
to  do  with  the  amount. 

55. — The  like  course  of  reasoning  will  give  us  the  result. 
Take  the  four  numbers  representing  the  present  worths  of  $1 .  00 
at  4,  3,  2  and  1  periods  respectively,  and  multiply  each  by  1.03. 
a.  b. 

Present  Worth  of  Present  Worth  of 

Annuity  of  |1.00  Annuity  of  $1.03 

.888487  .915142 

.915142  .942596   : 

.942596  .970874 

.970874  1.000000 

Canceling  all  equivalents,  we  have 
.888487  ....... 


1.000000 


Present  Worth  of 

Annuity  of  .03 

1.000000 

less         .888487 

.111513 


Annuity  of  $1 .  00  =  .  111513  -^  .  03  =      3 .  71710 

This  is  the  same  result  (rounded  up)  as  that  obtained  by  adding 

column  a, 

56.— But  .  111513  is  the  compound  discount  of  $1 .  00  for 
four  periods,  and  we  therefore  construct  this  rule : 

57. — Rule.  To  find  the  present  worth  of  an  annuity  of 
$1.00  for  a  given  time  and  rate,  divide  the  compound  discount 
for  that  time  and  rate  by  a  single  interest.  Symbolically 
P  =  D  -f-  /.  We  might  give  this  the  form  P  =  D  x  ^,  but  in 
practice  this  would  not  be  so  convenient. 

58. — It  may  assist  in  acquiring  a  clear  idea  of  the  working 
of  an  annuity,  if  we  anal3^se  a  series  of  annuity  payments 
from  the  point  of  view  of  the  purchaser. 


28  Thb  Accountancy  of  Investment. 

59.— He  who  invests  $3.7171  at  3%,  in  an  annuity  of  4 
periods,  expects  to  receive  at  each  payment,  besides  3%  on  his 
principal  to  date,  a  portion  of  that  principal,  and  thus  to  have 
his  entire  principal  gradually  repaid. 

Principal. 

60. — His  original  principal  is 3 .  7171 

At  the  end  of  the  first  period  he  receives  1 .  00 

consisting  of  3%  on  3.7171 1115 

and  payment  on  principal 8885  .8885 

leaving  new  principal 2 .  8286 

(or  present  worth  at  3  periods). 

In  the  next  instalment 1 .00 

there  is  interest  on  2.8286 0849 

and  payment  on  principal 9151  .9151 

leaving  new  principal 1 .  9135 

Third  instalment 1.00 

Interest 0574 

on  principal .9426  .9426 

.9709 

I^ast  instalment , 1 .  00 

Interest 0291 

Principal  in  full .9709  .9709 

61. — Thus  the  annuitant  has  received  interest  in  full  on 
the  principal  outstanding,  and  has  also  received  the  entire 
original  principal.  The  correctness  of  the  basis  on  which  we 
have  been  working  is  corroborated. 

62. — It  is  usual  to  form  a  schedule  showing  the  com- 
ponents of  each  instalment  in  tabular  form. 

Date  '^°^^^  Interest         Payments  principal 

Instalment       Payments        principal        Outstanding 

1904  Mar.  9  3.7171 

1904  Sept.  9      1.00     .1115     .8885     2.8286 

1905  Mar.  9       1.00     .0849     .9151     1.9135 

1905  Sept.  9      1.00     .0574     .9426     0.9709 

1906  Mar.  1       1.00     .0291     .9709     0.0000 

4.00     .2829    3.7171 

Had  the  purchaser  re-invested  each  instalment  at  3%,  he  would 
have,   at  the  end,  $4.1836  (Article  45),  which  is  equivalent 


Present  Worth  oe  an  Annuity.  29 

to  his  original  $3.7171  compounded  (3.7171  X  1.1255  = 
4.1836). 

63. — In  Article  43,  using  logarithms,  we  found  the  present 
worth  of  a  single  $1 .  00  at  50  periods,  at  3%,  to  be       .  2281071 

Subtracting  this  from 1.0000000 

we  have 7718929 

which  is  the  compound  discount.     Dividing  this 

by  .03  we  have 25 .72976 

which  is  the  present  worth  of  an  annuity  of  $1 .  00  for  50  periods. 
Thus  we  see  that  the  process  of  finding  the  present  worth  of 
an  annuity,  or,  as  it  is  termed,  evaluation,  is  rendered  very 
easy,  no  matter  how  long  the  time,  by  using  logarithms. 

64. — The  present  worth  of  an  annuity  of  $1.00  is  some- 
times called  the  number  of  years'  purchase.  Thus  we  would 
say,  in  the  example  just  given,  that  a  50  year  annuity,  at  3%, 
is  worth  nearly  26  years'  purchase ;  meaning  that  one  should 
pay  now  nearly  26  times  a  year's  income,  whatever  that  may 
be.  In  Hardcas tie's  "  Accounts  of  Executors,"  page  27  and 
following,  will  be  found  several  examples  of  the  evaluation  of 
leases  for  years,  which  are  a  species  of  annuity.  It  will  be 
useful  to  work  these  out  by  logarithms  to  as  many  places  as 
possible. 


30  The  Accountancy  of  Investment. 


CHAPTER  VI. 

Rent  of  Annuity  and  Sinking  Fund. 

65. — The  number  of  dollars  in  each  separate  payment  of 
an  annuity  is  called  the  rent  of  the  annuity. 

66. — We  saw  that  3.7171  is  the  present  worth  of  an  annu- 
ity composed  of  payments  of  1.00  each.  We  may  reverse  this 
and  say  that  1 .  00  is  the  rent  of  3 .  7171  invested  in  an  annuity  of 
4  payments  at  3%.  What,  then,  is  the  rent  to  be  obtained  by 
investing  $1  in  the  same  way  ?  Since  the  present  worth  has 
been  reduced  in  the  ratio  of  3.7171  to  1,  evidently  the  rent 
must  be  reduced  in  the  same  ratio,  that  is  1  -^-  3.7171.  By 
ordinary  division  or  by  logarithms,  this  quotient  is  .26903. 
Therefore  .26903  is  the  rent  of  an  annuity  of  4  terms  at  3%, 
for  every  $1  invested.  Or  $1  is  the  present  worth  at  3%  of  an 
annuity  of  .  26903.  This  may  be  illustrated  by  making  up  a 
schedule  : 

Rent. 

Beginning  of  first  period 

End  of  first  period 26903 

End  of  second  period 26903 

End  of  third  period 26903 

End  of  fourth  period 26903 

1.07612        .07612      1.00000 

67. — Rule,  To  find  the  rent  of  an  annuity  valued  at  $1, 
divide  $1  by  the  present  worth  of  an  annuity  of  $1  for  the  given 
rate  and  time.     Rent  =  1  -^-  P. 

68. — This  may  be  also  called  finding  how  much  per  period 
for  n  periods  at  the  rate  t  can  be  bought  for  $1.  A  borrower  may 
agree  to  pay  back  a  loan  in  instalments,  which  comprise  prin- 
cipal and  interest.  Suppose  a  loan  of  $1,000  were  made  under 
the  agreement  that  such  a  uniform  sum  should  be  paid  annually 
as  would  pay  off  (amortise)  the  entire  debt  with  3%  interest 
in  4  years.  The  present  worth  is,  of  course,  $1,000,  and  by  the 
above  process  each  instalment  or  contribution  would  be  $269 .  03. 
In  countries  imposing  an  income-tax  it  is  usual  to  incorporate 
in  the  bond  a  schedule  with  columns  like  those  in  Article  66, 
showing  what  part  of  the  instalment  is  interest,  as  that  alone 
is  taxable. 


TEREST. 

Reduction. 

Value. 
1.00000 

.03 

.23903 

.76097 

.02283 

.24620 

.51477 

.01544 

.25359 

.26118 

.00785 

.26118 

0. 

Rent  of  Annuity  and  Sinking  Fund.  31 


Annual  ^%^^^^^  Payment  principal 

Balance  Principal 


Instalment         «Jil^^  vri^nir^at         Outstanding 


Jan.    1  1901 1,000.00 

Dec.  31  1904  269.03  30.00  239.03  760.97 

Dec.  31  1905  269.03  22.83  246.20  514.77 

Dec.  31  1906  269.03  15.44  253.59  261.18 

Dec.  31  1907  269.03  7.85  261.18  0. 

1076.12  76.12        1,000.00 


69. — It  may  be  required,  also,  to  find  such  an  annuity  as 
will,  at  the  end  of  a  certain  number  of  periods,  have  accu- 
mulated to  $1.00  or  any  other  vSum^  This  is  called  a  sinking 
fund,  when  it  is  intended  to  provide  for  a  liability  not  yet 
matured.  In  the  case  exhibited  in  the  schedule,  in  Article  68, 
the  debt  was  amortised,  with  the  assent  of  the  creditor,  by 
gradual  payments.  Let  us  suppose  that  the  creditor  prefers  to 
wait  till  the  day  of  maturity,  and  receive  his  $1,000  at  once. 
He  must  be  paid  his  interest  of  $30  each  year,  but  the  debtor, 
to  provide  for  the  principal,  must  also  transfer  from  his  general 
assets  to  a  special  account  (or  into  the  hands  of  a  trustee,  if  he 
doubts  his  self-control),  where  it  will  draw  interest  at  S%,  such 
a  sum  as  will  accumulate  to  $1,000.  This  is  the  sinldng  fund. 
Since  $1.00,  set  aside^  annually ..  amounts,,  after  4  years,  to 
$4.183627,  to  find  what  sum  will  amount  to  $1,000,  we  must 
divide  1,000  by  4.183627,  giving  for  the  contribution  to  the 
sinking  fund  $239.03. 

70.— Rule.  To  find  what  annuity  will  amount  to  $1.00, 
or  what  should  be  each  sinking  fund  contribution  to  provide 
for  $1.00:  divide  $1.00  by  the  amount  of  an  annuity.  Sinking 
fund  contribution  ^  1  -^  S. 

71. — If  you  observe  the  two  results  obtained  by  the  two 
preceding  rules:  .26903 

and     .23903     you   will    see   that    they 
differ  by  .  03      ,  which  is  exactly  the  period- 

ical interest  on  the  original  loan.  Hence  the  amount  paid  in 
the  second  case,  if  interest  be  included,  is  just  the  same  as  in 
the  former.  This  is  as  it  should  be,  for  in  the  latter  case  we 
are  investing  in  some  other  3%  security,  while  in  the  former, 
we  are  investing  in  part  of  this  very  obligation.  Gradual  pay- 
ment, or  gradual  accumulation  for  a  single  payment,  come  to 
the  same  thing.  These  two  forms  of  practically  the  same  pro- 
cess, amortisation  and  sinking  fund,  will  be  useful  to  guide  us 
when  we  study  the  subject  of  premiums  on  securities. 


32  Thk  Accountancy,  of  Investmtjnt. 


CHAPTER  VII. 

NoMiNAi,  AND  Effective  Rates. 

72. — We  have  reduced  all  our  operations  to  so  many 
periods,  and  such  a  rate  per  period,  but  it  is  usual  to  speak  of 
such  a  rate  per  aiinum,  payable  so  many  times  a  year,  or  ' '  con- 
vertible half  yearly  or  quarterly."  Where  the  interest  is  pay- 
able otherwise  than  annually,  the  rate  per  annum  is  only 
nominally  correct.  For  example:  if  we  take  3%  per  half  year, 
this  would  be  nominally  6%  per  annum,  but  efFectively  it 
would  be  6.09%  per  annum,  because  1.03  X  1.03  =  1.0609. 
If  paid  quarterly,  the  effective  rate  per  annum,  6 .  1364%  would 
correspond  to  the  nominal  rate  6%.  Evidently,  the  more  fre- 
quent conversion  results  in  more  rapid  accumulation.  If  paid 
monthly,  the  effective  rate  would  be, 6. 1678%,  while  if  paid 
daily  it  would  be  6. 1826%.  But  there  is  a  limit  beyond  which 
this  acceleration  will  not  go ;  6%  compounded  every  minute, 
or  every  second,  or  every  millionth  of  a  second,  or  constantly, 
could  never  be  so  great  as  6 .  184%. 

73. — The  process  of  finding  the  effective  rate  follows 
naturally  from  the  ordinary  rule  for  compound  interest.  If 
the  nominal  rate  is  6%  per  annum,  and  it  be  paid  quarterly, 
the  actual  ratio  is  1.015,  and  the  fourth  power  of  1.015  is  the 
amount  at  the  end  of  the  fourth  quarter,  which  by  multiplica- 
tion or  by  adding  logarithms  is  found  to  be  1 .  061364,  or  interest 
.  061364.  It  will  well  exemplify  the  logarithmic  process  if  we 
apply  it,  finding  an  effective  rate  for  daily  compounding.  Let 
the  nominal  interest  be  6%,  then  the  actual  rate  per  period 
of  a  day  will  be  .  06  -f-  365.  We  will  first  perform  this  division 
by  logarithms: 

Ivog.  .06  2.7781513 

—  Log.  365  2.5622929 

Difference  4.2158584 

4.2158584  is,  we  find,  the  logarithm  of  .0001643835,  hence  the 
daily  ratio  of  decrease  is  1.0001643835;  and  we  again  find  the 
logarithm  of  this  number,  which  is  .00007138;  365  times 
.00007138  being    .02605370,   which  is  opposite  the  number 


NOMINAI.   AND   KfFKCTIVK   RATKS.  33 

1.061826.     If  m  represent  the  number  of  payments  per  year, 
andy  the  effective  rate,  we  have/  =  (1  +  ^)  *^  —  1. 

74. — It  may  also  be  necessary  to  solve  another  problem, 
in  order  to  produce  an  effective  rate  of  6%  per  annum,  what 
nominal  rate  is  required,  conversion  being  half-yearly,  quarterly, 
etc.  In  this  case  we  have  to  find  the  ratio  of  increase  for  the 
lesser  unit  of  time,  which  we  do  by  dividing  the  logarithm  of 
the  effective  rate  by  the  number  of  conversions.  Thus,  if  we 
are  required  to  find  the  nominal  rate,  which,  compounded 
quarterly,    will   be   equivalent   to   the  effective   rate  6%,  we 

divide  the  logarithm  of  1.06 02530587 

by  4,  giving 00632647 

The  number  opposite  this  is ... .    1 .  0146738 

This  being  the  quarterly  ratio,  the  nominal  annual 

rate  will  be  4  times  this  rate 0586952 

75. — There  are  some  other  problems  in  compound  interest, 
such  as  finding  the  time  or  the  rate,  when  the  other  elements 
are  given.  But  the  foregoing  rules  will  suffice  for  most  of  the 
purposes  of  investment  accounts. 


34  ^HS  Accountancy  of  Investment. 


CHAPTER  VIII. 
Vai^uation  of  Bonds. 

76. — Investments  of  loan-capital  are  usually  made  by 
means  of  written  instruments,  known  as  debentures  or  bonds. 
These  are  promises  to  pay:  first,  a  principal  sum  at  a  certain 
date  in  the  future ;  this  principal  sum  is  the  par  value  of  the 
bond;  secondly,  to  pay  at  the  end  of  each  period,  as  interest,  a 
certain  percentage  of  the  principal.  The  bond  also  contains 
provisions  as  to  the  time,  place  and  manner  of  these  payments, 
and  usually  refers,  also,  to  the  security  obligated  to  insure  its 
fulfillment,  and  to  the  law  (in  case  of  a  corporation,  public  or 
private)  which  authorizes  the  issue.  In  Prof.  Frederick  A. 
Cleveland's  "Funds  and  their  Uses"  will  be  found  furthei 
particulars  as  to  the  various  descriptions  of  bonds  and  similar 
securities. 

77. — The  rate  of  interest  named  in  the  bond  is  usually  aD 
integer  per  cent.,  or  midway  between  two  integers:  as  2%, 
2>^%,  3%,  3^%,  4%,  4^,  5%,  6%,  7%.  Occasionally,  such 
odd  rates  occur,  as  3}(%,  3.60%,  3.65%,  3%%,  but  these  are 
unusual  and  inconvenient.  Most  bonds  pay  interest  semi- 
annually, and  on  the  first  day  of  the  month.  Here  again  there 
is  some  deviation.  A  considerable  number  of  issues  pay  interest 
quarterly,  and  a  very  few  annually.  A  very  few  have  the 
interest  fall  due  on  some  other  day  than  the  first  of  the  month. 
These  vagaries  are  of  no  benefit,  and  slightly  injure  the  value 
of  the  bond.  It  would  in  some  respects  be  better,  however,  if 
interest  were  payable  on  the  lasi  day  of  the  half  year,  thus 
bringing  the  entire  transaction  inside  of  a  calendar  period. 

78. — Bonds  are  usually  designated  according  to  the  obligor, 
the  rate  of  interest,  the  date  or  year  of  maturity,  adding,  if 
requisite,  the  initials  of  the  months  when  interest  is  payable. 
Thus,  "  Manhattan  4's  of  '90,  J  &  J,"  indicates  the  bonds  of 
the  Manhattan  Railway  Company,  bearing  4%  interest  per 
annum,  payable  semi-annually  on  the  first  days  of  January 
and  July,  and  payable  in  1990. 


VAI.UATION  OF  Bonds.  35 

79. — Bonds  very  frequently  are  bought  and  sold  at  a  differ- 
ent price  from  par.  This  has  its  effect  on  the  income  derived 
from  the  investment.  The  amount  invested  being  greater,  the 
percentage  of  fixed  income  is  less;  beside  this,  the  excess  or 
premium  will  not  be  repaid  at  maturity,  but  will  be  sacrificed; 
hence,  a  bond  purchased  above  par  earns  less  than  the  con- 
tractual interest.  Similarly,  if  the  purchase  is  below  par,  the 
percentage  of  fixed  income  is  greater;  besides,  at  maturity  the 
owner  will  receive  not  only  all  that  he  invested,  but  also  the 
discount  bringing  it  up  to  par.  Hence  a  bond  purchased  below 
par  earns  more  than  the  contractual  interest. 

80. — ^There  are  thus  two  rates  of  interest  relatively  to  the 
par  and  the  price :  a  nominal  rate,  which  is  so  many  hundredths 
of  par;  and  an  effective  rate,  which  is  so  many  hundredths  of 
the  amount  invested  and  remaining  invested.  The  words, 
nominal  and  effective,  are  as  correctly  applied  in  this  case  as  in 
relation  to  frequency  of  conversion;  but  for  the  sake  of  dis- 
tinction we  shall  prefer  to  call  them  the  cash  rate  and  the 
income  rate,  and  designate  them,  when  desirable,  by  the 
symbols  c  and  i  respectively.  1  +  /  is  the  ratio  of  increase  as 
heretofore.  1  +  ^  is  not  required,  as  c  is  not  an  accumulative 
rate,  but  merely  an  annuity  purchased  with  the  bond.  The 
difference  of  rates  is  c  —  /  or  i  —  c. 

81. — The  following  are  some  of  the  expressions  used  to 
denote  an  investment  made  above  or  below  par :  * '  6%  bond 
to  net  5%;"  ''6%  bond  on  5%  basis;"  "6%  bond  yielding 
5%;"   '' 6%  bond  pays  5%. " 

82. — In  a  bond  purchased  above  or  below  par,  we  have, 
therefore,  the  following  elements:  the  par  principal  payable 
after  n  periods  ;  an  annuity  of  c  per  cent,  of  par  for  71  periods, 
and  a  ratio  of  increase,  1  -j-  i.  Given  these,  there  are  two  dis- 
tinct methods  for  finding  the  value  of  the  entire  security,  and 
these  must  give  the  same  result. 

83. — First  Method.  Separate  Evaluation  of  Principal 
and  Annuity.  Let  us  suppose  a  1%  bond,  interest  semi- 
annually, 25  years  to  run  (50  periods),  for  $1,000.  The  present 
value  is  composed  of  two  parts:  (1)  the  present  worth  of  $1,000 
in  a  single  payment,  50  periods  hence;  (2)  an  annuity  of  $35 


36  Thk  Accountancy  of  Investment. 

for  50  terms.  We  can  only  value  these  when  we  know  what  is 
the  income-rate  current  upon  securities  of  this  grade.  Let  us 
assume  3V  as  the  income- rate  per  period,  or  what  is  usually 
called  a  6%  basis.     The  ratio  of  increase  is  1.03. 

84. — The  first  part  of  the  solution  is  to  find  the  present 
worth  of  $1,000  at  3%  in  50  periods.  In  Article  42,  we  have 
found  the  present  worth  of  $1.00  on  the  same  conditions, 
which  is  .  2281071;  hence,  the  value  of  the  $1,000  is  $228 .  1071. 
It  will  be  noticed  that  this  result  has  not  the  slightest  reference 
to  the  1%  rate  of  the  bond.  As  a  matter  of  compound  interest, 
the  1%  does  not  exist. 

85. — We  next  have  to  value  an  annuity  of  50  terms  at  $35. 
In  Article  63,  we  valued  a  similar  annuity  of  $1.00  and 
found  it  to  be  worth  $25.72976.  If  each  term  be  $35,  the 
value  will  be  $900 .5417.  Adding  this  to  the  value  of  the  $1,000, 
we  have  the  value  of  the  bond,  228.1071  +900.5417=$1128.6488. 
The  ordinary  tables,  which  give  the  values  of  a  $100  bond  only, 
read  112 .  86,  which  is  the  same,  rounded  off.  The  above  com- 
putation gives  a  result  which  is  correct  to  the  nearest  cent  on 
$100,000,  viz.,  $112,864.88. 

86. — Second  Method.  Division  of  Income  and  Eval- 
uation of  Premium  or  Discount.  Each  semi-annual  pay- 
ment of  $35  may  be  divided  into  two  parts:  $30  and  $5.  The 
$30  is  the  3%  income  on  the  $1,000;  we  may  disregard  this 
and  consider  only  the  $5,  which  is  surplus  interest,  and,  in 
fact,  is  an  annuity  which  must  be  paid  for  in  a  premium. 
Having  devoted  $30  to  the  payment  of  our  expected  income- 
rate  on  par,  we  have  $5,  the  difference  of  rates  per  period,  as  a 
benefit  to  be  valued. 

87. — We  have  found  the  present  value  of  an  annuity  of 
$1 .  00  to  be  25 .  72976.  Multiplying  this  by  5  to  get  the  present 
worth  of  a  $5  annuity,  we  have  $128. 6488,  which  is  the  premium, 
agreeing  with  the  result  of  the  previous  method.  The  second 
method  is  not  only  quicker,  but  it  often  gives  one  more  place  of 
decimals. 

88. — In  the  case  of  a  bond  sold  below  par,  the  cash-rate 
being  less  than  the  income-rate,  the  same  procedure  is  followed 
for  finding  the  present  worth  of  $5,  but  the  result,  $128.6488, 


Valuation  of  Bonds.  37 

is  subtracted  from  the  par,  giving  $871 .  3512  as  the  value  of  a 
5%  bond  earning  6%  per  annum. 

89. — As  this  second  method  is  superior  to  the  first,  we  will 
adopt  it  as  the  standard. 

90. — Rule.  The  premium  or  discount  on  a  bond  for  $1 .  00 
bought  above  or  below  par,  is  the  present  worth  of  an  annuity 
of  the  difference  of  rates. 

91.— We  have  found  the  value  of  a  7%  bond  for  $100, 
paying  6%  (semi-annual),  25  years  to  run,  to  be  $1128.65  to 
the  nearest  cent.  This  is  the  amount  which  must  be  invested 
if  the  6%  income  is  to  be  secured.  At  the  end  of  the  same 
half  year,  the  holder  must  receive  3%  interest  on  this  $1128 .  65, 
which  is  $33 .86.  But  he  actually  collects  $35,  and  after  using 
$33.86  as  revenue,  he  must  apply  the  remainder,  $1.14,  to  the 
amortisation  of  the  premium.  This  will  leave  the  value  of  the 
bond  at  the  same  income-rate,  $1127.51.  If  our  operations 
have  been  correct,  the  value  of  a  7%  bond  to  net  6%,  24:}4 
years  or  49  periods  to  run,  will  be  $1127 .51.  To  test  this,  and 
to  exemplify  the  method,  we  will  go  through  the  entire 
operation: 

92.— The  logarithm  of  1.03  is 01283722 

This  X  49  = 6290238 

This  subtracted  from  0  = 1.3709762 

We  find  that  the  logarithm  of  .23495  is 1.3709754 

Remainder 8 

which  gives  the  figures  02. 
Hence,  .  2349502  is  the  total  discount  at  3%  per  period  for  49 
periods  on  $1.00.  Subtracting  the  .2349502  from  1,  we  have 
.7650498,  which  is  the  present  value  of  an  annuity  of  .03  for 
49  periods.  Dividing  by  .  03  we  have  the  present  value  of  an 
annuity  of  $1.00  per  period,  viz.,  25.50166.  But  the  surplus 
interest  (35  —  30)  is  $5;  hence,  we  must  multiply  25.50166  by 
5,  giving  $127,508,  or,  rounded  off,  $127.51,  as  the  premium, 
at  49  periods.  Adding  this  to  the  1,000  we  have  $1127 .51,  the 
same  result  as  in  Article  91. 

93. — When  bonds  are  purchased  as  investment,  a  Schedule' 
of  Amortisation  should  be  constructed,  showing  the  gradual 
extinction  of  the  premium  by  the  application  of  surplus  interest. 


38  Th^  Accountancy  of  Investment. 

The  following  is  the  form  recommended,  but  it  should  be  con- 
tinued to  the  date  of  maturity,  and  at  intervals  corrected  in  the 
last  figure  by  a  fresh  logarithmic  computation. 

SCHEDUI.E  OF  Amortisation. 

7%  bond  of  the ,  payable  Jan.  1,  1954.     Net 

6%.     J  J. 

Total  Net  ^^  , 

Date  Interest       Income    Amortisation     f.„,,_  Par 

7%  6%  V^l"« 

1904  Jan.  1     Cost 1,128.65  1,000.00 

July  1  35.00    33.86    1.14    1,127.51 

1905  Jan.  1  35.00    33.83    1  17    1,126.34 
July  1  35.00    33.79    1.21    1,125.13 

etc.,  etc.,  etc. 

' '  Book  Value  ' '  might  also  be  termed  *  *  Investment  Value. ' ' 

94. — This  schedule  is  the  source  of  the  entry  to  be  made 
each  half  year  for  "writing  off"  or  "writing  up"  the  premium 
or  discount,  so  that  at  maturity  the  bond  will  stand  exactly  at 
par.  We  give  two  more  examples  continuing  them  to  maturity, 
one  being  above  par  and  the  other  below  par.  As  the  formation 
of  schedules  is  the  basis  of  the  accountancy  of  amortised 
securities,  we  .shall  present  the  same  materials  in  various  forms, 
lettering  them  (A),  (B),  etc. 

SCHEDUI.E  OF  Amortisation  (A). 
5%  Bond  of  the ,  payable  May  1,  1909.     M  N. 


Total  Net 


Date  Interest        Income       Amortisation  tt„i..^  Par 


Book 


5%  4% 


Value 


1904  May^l  ^                                      Cost  104,491.29       100,000.00 
Nov. 'l  2,500  2,089.83  410.17  104,081.12 

1905  May  1  2,500  2,081.62  418.38  103,662.74 
Nov.  1  2,500  2,073.26  426.74  103,236.00 

1906  May  1  2,500  2,064.72  435.28  102,800.72 
Nov.  1  2,500  2,056.01  443.99  102,356.73 

1907  May  1  2,500  2,047.13  452.87  101,903.86 
Nov.  1  2,500  2,038.08  461.92     .    101,441.94 

1908  May  1  2,500  2,028.84  471.16  100,970.78 
Nov.  1  2,500  2,019.42  480.58  100,490.20 

1909  May  1  2,500  2,009.80  490.20  100,000.00 

25,000  20,508.71  4,49129 


Valuation  of  Bonds.  39 

Schedule  of  Accumulation  (B). 
3%  Bond  of  the ,  payable  May  1, 1909.     M  N. 


Total  Net 


Book 


Interest        Income      Accumulation  vaii,* 

.    3%  4%  ^^^"* 


1904  May  1  Cost  95,508.71      100,000.00 
Nov.  1  1,500  1,910.17  410.17  95,918.88 

1905  May  1  1,500  1,918.38  418.38  96,337.26 
Nov.  1  1,500  1,926.74  426.74  96,764.00 

1906  May  1  1,500  1,935.28  435.28  97,199.28 
Nov.  1  1,500  1,943.99  443.99  97,643.27 

1907  May  1  1,500  1,952.87  452.87  98,096.14 
Nov.  1  1,500  1,961.92  461.92  98,558.06 

1908  May  1  1,500  1,971.16  471.16  99,029.22 
Nov.  1  1,500  1,980.58  480.58  99,509.80 

1909  May  1  1,500  1,990.20  490.20  100,000.00 


15,000      19,491.29        4,491.29 


.9;i. — It  will  be  observed  in  these  two  schedules  that  the 
one  is  exactly  as  much  above  par  as  the  other  is  below  it,  and 
that  the  ' '  accumulation ' '  and  ' '  amortisation ' '  are  exactly  the 
same  in  both,  being  added  in  one  case  and  subtracted  in  the 
other.  In  one  line  the  net  income  is  apparently  in  error  1  cent, 
but  this  is  on  account  of  the  roundings  of  the  fractions  of  a 
cent,  and  would  disappear  if  the  operation  were  carried  to  one 
place  further. 

95. — The  figures  in  the  column  "  Book  Value  "  might  be 
taken  from  the  tables  of  bond  values,  published  in  book  form. 
The  column  of  amortisation  would,  in  this  case,  be  derived 
from  the  Book  Values,  and  the  Net  Income  from  the  Amorti- 
sation. The  schedule  would  then  be  roughly  accurate,  unless 
the  table  used  were  carried  to  a  greater  number  of  places  than 
is  usually  done.  Sprague's  Bond  Tables  will  give  eight  places 
instead  of  four,  and  from  them  schedules  (A)  and  (B)  can  be 
obtained  to  the  nearest  cent. 


40                 1 

rni 

J  Accountancy  of  : 

Invkstment. 

(C) 

Total 

Net 

Book 

Date 

Interest 

Income 

Amortisation 

Value 

5% 

^% 

Approximate 

1904  May 

104,490 

Nov. 

2,500 

2,090 

410 

104,080 

1905  May 

2,500 

2,080 

420 

103,660 

Nov. 

2,500 

2,080 

420 

103,240 

1906   May 

2,500 

2,060 

440 

102,800 

Nov. 

2,500 

2,060 

440 

102,360 

1907  May 

2,500 

2,040 

460 

101,900 

Nov. 

2,500 

2,040 

460 

101,440 

1908  May 

2,500 

2,030 

470 

100,970 

Nov. 

2,500 

2,020 

480 

100,490 

1909   May 

2,500 

2,010 

490 

100,000 

96. — It  will  be  observed  that  in  schedules  (A)  and  (B) 
the  entire  interest  is  accounted  for,  both  the  interest  on  the 
par  and  that  on  the  premium.  We  may  easily  construct  the 
schedule  so  as  to  eliminate  the  par  and  its  interest  at  the  rate  i, 
and  deal  only  with  the  surplus  interest  or  the  deficient  interest, 
according  to  the  theory  in  Article  86.  As  this  may  be  prefer- 
able for  some  forms  of  accounts,  we  again  work  out  the  schedule 
for  ''5%  bond  net  4%,  5  years,  semi-annual": 

(D) 


Surplus 

Interest  on 

Date 

Interest 
1% 

Premium 

Amortisation 

Premium 

1904  May 

4,491.29 

Nov. 

500 

89.83 

410.17 

4,081.12 

1905  May 

500 

81.62 

418.38 

3,662.74 

Nov. 

500 

73.26 

426.74 

3,236.00 

1906   May 

500 

64.72 

435.28 

2,800.72 

Nov. 

500 

56.01 

443.99 

2,356.73 

1907   May 

500 

47.13 

452.87 

1,903.86 

.      Nov. 

500 

38.08 

461.92 

1,441.94 

1908   May 

500 

28.84 

471.16 

970.78 

Nov. 

500 

19.42 

480.58 

490.20 

1909   May 

500 

9.80 

490.20 

0 

5,000 

508.71 

4,491.29 

VAI.UATION  OF  Bonds.  41 

■97. — We  have  hitherto  assumed  that  the  purchase  of  the 
bond  took  place  exactly  upon  an  interest  date.  We  must  now 
consider  the  case  when  the  initial  date  differs  from  the  interest 
date.  I^et  us  suppose  the  purchase  to  take  place  on  July  1,  when 
one-third  of  the  period  has  elapsed.  The  business  custom  is  to 
adjust  the  matter  as  follows:  The  buyer  pays  to  the  seller  the 
(simple)  interest  accrued  for  the  two  months,  acquiring  thereby 
the  full  interest -rights,  which  will  fall  due  on  November  1,  and 
the  premium  is  also  considered  as  vanishing  by  an  equal  portion 
each  day,  so  that  one- third  of  the  half-yearly  amortisation  takes 
place  by  July  1.  The  amortisation  from  May  1  to  November 
1  being  $410.17,  that  from  May  1  to  July  1  must  be  $136.72, 
and  the  book  value  on  July  1  is  $104,354.57,  with  accrued 
interest,  $833 .  33— in  all  $105,187 .  90.  This  last  number  is  the 
flat  price,  that  is  to  say,  it  is  inclusive  of  interest.  It  might 
have  been  obtained  in  the  following  manner : 

To  the  value  on  May  1 $104,491.29 

add  simple  interest  thereon,  at  4%,  for  2  months . ,  696 .  61 

giving  the  flat  price $105,187.90 

In  buying  bonds,  there  is  usually  a  stipulation  that  the  price 
should  be  so  many  per  cent.  * '  and  interest, ' '  otherwise  the  price 
named  is  understood  to  be  "  flat. ' ' 

98. — This  practice  of  adjusting  the  price  at  intermediate 
dates  by  simple  interest  is  conventionally  correct,  but  is  scien- 
tifically inaccurate,  and  always  works  a  slight  injustice  to  the 
buyer.  The  seller  is  having  his  interest  compounded  at  the  end 
of  two  months  instead  of  six  months,  and  receives  a  benefit 
therefrom,  at  the  expense  of  the  buyer.  It  will  readily  be 
seen  that  the  buyer  does  not  net  the  effective  rate  of  4%  semi- 
annually on  his  investment  of  $105,187.90.  The  true  price 
would  be  $105,183.31,  giving  both  buyer  and  seller,  not  4% 
nominal,  but  the  equivalent  effective  with  bi-monthly  and  four- 
monthly  conversion.  In  practice,  however,  for  any  time  above 
six  months,  simple  interest  is  generally  used,  to  the  slight  dis- 
advantage of  the  buyer,  who  may  claim,  and  probably  legally, 
that  the  November  value  -f  interest  due  should  have  been  dis- 
counted at  4%;  106,581.12-^1.01/3  ==105,178.74;  which  is  al- 
most exactly  as  much  too  low  as  the  $105,187.90  is  too  high. 

99. — The  schedule  would,  therefore,  in  practice,  read  as 
follows : 


42  Thb  Accountancy  of  Investment. 

(E) 


Total  Net 

Date  Interest  Income      Amortisation 

5%  i% 


Book 
Value 


1904  July  1                                                     Cost  104,354.57     100,000.00 
Nov.  1  1,666.67  1.393.22  273.45  104,081.12 

1905  May  1  2,500.00  2,081.62  418.38  103,662.74 
Nov.  1  2,500.00  2,073.26  426.74  103,236.00 

1906  May  1  2.500.00  2,064.72  435.28  102,800.72 
Nov.  1  2,500.00  2,056.01  443.99  102,356.73 

1907  May  1  2,500.00  2.047.13  452.87  101,903.86 
Nov.  1  2,500.00  2,038.08  461.92  101,441.94 

1908  May  1  2,500.00  2,028.84  471.16  100,970.78 
Nov.  1  2.500.00  2,019.42  480.58  100,490.20 

1909  May  1  2,500.00  2,009.80  490.20  100,000.00 


24,166.67      19.812.10     4,354.57 


100. — The  interest  dates  may  not  always  be  the  most  con- 
venient epochs  for  periodical  valuation.  There  may  be  many 
kinds  of  bonds,  the  interest  on  some  falling  due  in  every  rponth 
in  the  year,  and  yet  on  a  certain  annual  or  semi-annual  date 
the  entire  holdings  must  be  simultaneously  valued.  It  will 
then  be  convenient  if  we  can  arrange  our  schedules  so  that 
without  recalculation  every  book  value  will  be  ready  to  place 
in  the  balance  sheet.  Fortunately,  this  is  easier  than  would 
be  supposed. 

101. — We  again  take  a  5%  bond,  payable  on  Nov.  1, 1904, 
on  a  4%  basis,  but  we  assume  that  the  investor  closes  his  books 
on  the  last  days  of  June  and  December.  We  will  suppose  that 
the  purchase  is  made  on  August  1.  As  this  is  between  the 
May  and  the  November  periods,  we  must  adjust  the  price  as  in 
Article  96,  so  that  the  August  price  is  midway  between 
104,491.29  and  104,081.12,  namely:  $104,286.20  and  interest, 
being  the  customary,  not  the  theoretical,  method.  The  No- 
vember value  need  not  enter  into  the  schedule,  but  we  must 
locate  the  December  31  value,  just  as  we  found  the  July  1  value 
in  Article  96.  One-third  the  difference  between  $104,081.12 
and  $103,662.74,  or  $418.38,  is  $139.46;  104,081.12  —  139.46 
=  103,941.66.     Our  schedule  so  far  reads: 

Aug.    1  Cost  104,286.20 

Dec.  31         2,083.33  1,738.79  344.54  103,941.66 

Proceeding  in  the  same  way  to  find  the  value  on  June  30,  1905, 
from  those  of  May  1  and  November  1,  we  get  $103,520.49. 


Valuation  of  Bonds.  43 

'  102.— But  6  months  interest  at  4?^  on  $103,941.66  is 
$2,078.83,  which,  subtracted  from  $2,500,  gives  the  amorti- 
sation $421.17,  and  this,  written  off  from  $103,941.66,  gives 
$103,520.49,  precisely  the  same  as  obtained  by  interpolation 
between  May  and  November.  Hence  we  have  two  ways  of 
continuing  the  schedule  :  interpolation  and  multiplication.  In 
this  respect  the  commercial  practice  is  much  more  convenient 
than  the  theoretical  one.  Having  once  adjusted  the  value  at 
one  of  the  balancing  periods,  we  can  derive  all  the  remaining 
by  subtracting  the  net  income  from  the  cash  interest  and  re- 
ducing the  premium  by  the  difference,  completely  ignoring  the 
values  on  interest  days  (M  &  N). 

f/wn..  103. — No  diflBculty  arises  until  we  reach  the  broken  period, 
l^ttiy^l  —  May  1,  1909.  Here  the  computation  of  the  second 
column.  Net  Income,  is  peculiar.  The  par  and  the  premium 
must  be  treated  separately.  The  net  income  on  $100,000  is 
taken  at  fs  of  2%  for  the  fs  time,  giving  $1,333.33.  The 
premium,  $326.80,  however,  must  always  be  multiplied  by  the 
full  2%,  giving  $6 .  54.  Adding  $1,333 .  33  and  $6 .  54,  we  have 
$1,339.87,  which,  used  as  heretofore,  reduces  the  principal  to 
par.  The  reason  for  this  peculiarity  is  that  $490.20,  not 
$326.80,  is  the  conventional  premium,  on  which  4%  is  to  be 
computed;  hence,  instead  of  taking  3/2  of  $326.80  for  ^  of  a 
period,  we  take  $326 .  80  itself  for  a  whole  period,  two- thirds  of 
three-halves  being  unity. 

(F) 


Total 

Net 

Date 

Interest 
5% 

Income 
4% 

Amortisation 

Book                     p 
Value                     ^^^ 

1904   Aug.     1 

Cost 

104,286.20     100,000.00 

Dec.  31 

2,083.33 

1,738.79 

344.54 

103,941.66 

1905  June  30 

2,500.00 

2,078.83 

421.17 

103,520.49 

Dec.  31 

2,500.00 

2,070.41 

429.59 

103,090.90 

1906  June  30 

2,500.00 

2,061.82 

438.18 

102,652.72 

Dec.  31 

2,500.00 

2,053.05 

446.95 

102,205.77 

1907  June  30 

2,500.00 

2,044.12 

455.88 

101,749.89 

Dec.  31 

2,500.00 

2,035.00 

465.00 

101,284.89 

1908  June  30 

2,500.00 

2,025.70 

474.30 

100,810.59 

Dec.  31 

2,500.00 

2,016.21 

483.79 

100,326.80 

1909   May     1 

1,666.67 

1,339.87 

326.80 

100,000.00 

23,750.00 

19,463.80 

4,286.20 

44  Tii:Bi  Accountancy  of  Invkstme^nt. 

104. — In  all  the  foregoing  examples  it  has  been  assumed 
that  the  bond  has  been  bought  "on  a  basis,"  which  means 
that  the  buyer  and  seller  have  agreed  upon  the  income  rate 
which  the  bonds  shall  pay,  and  that  from  this  datum  the  price 
has  been  adjusted.  But  in  probably  the  majority  of  cases  the 
bargain  is  made  "  at  a  price,"  and  then  the  income  rate  must 
be  found.     This  is  a  more  difi&cult  problem. 

305. — The  best  method  of  ascertaining  the  basis,  when  the 
price  is  given,  is  by  trial  and  approximation  —  in  fact,  all 
methods  more  or  less  depend  upon  that.  The  ordinary  tables 
will  locate  several  figures  of  the  rate,  and  one  more  figure  can 
safely  be  added  by  simple  proportion.  But  it  is  an  important 
question  to  what  degree  of  fineness  we  should  try  to  attain. 
It  seems  to  be  the  consensus  of  opinion  and  practice  that 
to  carry  the  decimals  to  hundredths  of  one  per  cent,  is  far 
enough,  although  in  some  cases,  by  introducing  eighths  and 
sixteenths,  two-hundredths  and  four-hundredths  may  be  re- 
quired. Sprague's  Tables  give,  by  the  use  of  auxiliary  figures, 
values  for  each  one-hundredth  of  one  per  cent. 

106.— I.et  us  suppose  that  the  $100,000  5%  bonds,  5  years 
to  run,  MN,  are  ojffered  at  the  round  price  of  104>^  on  May  1, 
1904.  It  is  evident  that  this  is  nearly,  but  not  quite,  a  4^%  basis. 
Trying  a  3.99%  basis  we  find  that  the  premium  is  $4,537.39, 
which  is  further  from  the  price  than  is  $4,491 .  29,  the  4%  basis. 
Hence,  4%"  is  the  nearest  basis  within  ^^  of  one  per  cent.  In 
fact,  by  repeated  trials,  we  find  that  the  rate  is  about  .0399812 
per  annum.  It  is  manifest  that  such  a  ratio  of  increase  as 
1.0199906  would  be  very  unwieldy  and  impracticable,  and  that 
such  laborious  exactness  would  be  intolerable.  Yet  here  we 
have  paid  $104,500,  and  the  nearest  admissible  basis  gives 
$104,491.29;  what  shall  be  done  with  the  odd  $8.71  ?  It  must 
disappear  before  maturity,  and  on  a  4%  basis  it  will  be  even 
larger  at  maturity  than  now.  Three  ways  of  ridding  ourselves 
of  it  may  be  suggested. 

107. — First    Method   of  Eliminating  Residues.     Add 

the  residue  $8.71  to  the  first  amortisation,  thereby  reducing 
the  value  to  an  exact  4%  basis  at  once.  In  our  example  (A), 
instead  of  $410.17,  the  first  amortisation  would  be  $418.88. 


f  OF  THE     "^ 

\  OF 

VAI.UATION  OF  Bonds.  45 

This  is  at  the  income  rate  of  about  3 .  983%  for  the  first  half 
year  and  thereafter  at  4%.  For  short  bonds  the  result  is  fairly 
satisfactory. 

108.— Second  Method.  Divide  $8.71  into  as  many 
parts  as  there  are  periods.  This  would  give  .  87  for  each  period, 
except  the  first,  which  would  be  .88  on  account  of  the  odd 
cents.  Set  down  the  4:%  amortisation  in  one  column,  the  .87 
in  the  next,  and  the  adjusted  figures  in  the  third: 


410.17 

.88 

411.05 

418  38 

.87 

419.25 

426.74 

.87 

427.61 

435.28 

.87 

436.15 

443.99 

.87 

444.86 

452.87 

.87 

453.74 

461.92 

.87 

462.79 

471.16 

.87 

472.03 

480.58 

.87 

481.45 

490.20 

.87 

491.07 

The  following  will  then  be  the  schedule: 

(G) 


Date 

Total 
Interest 

Net 
Income 

Amort  isatiot 

Book                    p 
^         Value                    ^^^ 

5% 

*%  (-) 

1904  May  1 

104,500.00     100,000.00 

Nov.  1 

2,500.00 

2,088.95 

411.05 

104,088.95 

1905  May  1 

2,500.00 

2,080.75 

419.25 

103,669.70 

Nov.  1 

2,500.00 

2,072.39 

427.61 

103,242.09 

1906   May  1 

2,500.00 

2,063.85 

436.15 

102,805.94 

Nov.  1 

2,500.00 

2,055.14 

444.86 

102,361.08 

1907   May  1 

2,500.00 

2,046  26 

453.74 

101,907.34 

Nov.  1 

2,500.00 

2,037.21 

462.79 

101,444.55 

1908   May  1 

2,500.00 

2,027.97 

472.03 

100,972.52 

Nov.  1 

2,500.00 

2.018.55 

481.45 

100.491.07 

1909   May  1 

2.500.00 

2,008.93 

491.07 

100,000.00 

25,000.00 

20,500.00 

4,500.00 

109. — In  example  (G)  the  income  rate  varies  from  3 .99798 
to  3.99828;  hence  the  approximation  is  suflSciently  close  for 
any,  except  large  holdings  for  long  maturities. 

110. — Third  Method.  For  still  greater  accuracy,  we  may 
divide  the  $8 .  71  in  parts  proportionate  to  the  amortisation. 
The  amortisation  on  the  A%  basis  runs  off  $4,491.29,  and  we 
have  an  extra  amount  of  $8 .  71  to  exhaust.     Dividing  the  latter 


46 


The  Accountancy  of  Investment. 


by  the  former,  we  have  as  the  quotient  .  00194,  which  is  the 
portion  to  be  added  to  each  dollar  of  amortisation.  With  this 
we  form  a  little  table  for  the  9  digits: 

100194 
200388 
300582 
400776 
500970 
601164 
701358 
801552 
901746 

From  this  table  it  is  easy  to  adjust  each  item  of  amortisation, 
writing  down,  for  example,  to  the  nearest  mill: 


410 

.17 

.400 

.776 

10.019 

.100 

.070 

410.97 


418.38 


419.19 


426.74 


427.57 


435.28 


400.776 

400.776 

400.776 

10.019 

20.039 

30.058 

8.016 

6.012 

5.010 

.301 

.701 

.200 

.080 

.040 

.080 

436.12 


The  result,  in  schedule  (H),  varies  at  the  most  5  cents. 

(H) 


Total  Net 

Interest  Income      Amortisation 

6%  4%  (-) 


Book 
Value 


Par 


1904  May  1 
Nov.  1 

1905  May  1 
Nov.  1 

1906  May  1 
Nov.  1 

1907  May  1 
Nov.  1 

1908  May  1 
Nov.  1 

1909  May  1 


2,500.00 
2.500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 
2,500.00 


2,089.03 
2,080.81 
2,072.43 
2,063.88 
2,055.15 
2,046.25 
2,037.18 
2,027.93 
2,018.49 
2.008.85 


410.97 
419.19 
427.57 
436.12 
444.85 
453.75 
462.82 
472.07 
481.51 
491.15 


104,500.00 
104,089.03 
103,669.84 
103,242.27 
102,806.15 
102,361  30 
101,907.55 
101,444.73 
100,972.66 
100,491.15 
100,000.00 


25,000.00      20,500.00     4,500.00 


100,000.00 


111. — Short  Terminals.  It  sometimes  happens  (though 
infrequently)  that  the  principal  of  a  bond  is  payable,  not  at  an 
interest  date,  but  from  one  to  five  months  later,  making  a  short 
terminal  period.  The  author  has  discovered  a  very  simple 
method  of  obtaining  the  present  value  in  this  case.  It  will  not 
be  necessary  to  demonstrate  it,  but  an  example  will  test  it. 


Vai^uation  of  Bonds. 


47 


■  112. — Suppose  the  5%  bond,   M  N,  yielding  4%,  bought 

May  1,  1904,  were  payable  October  1,  instead  of  May  1,  1909, 

or  10|  periods.     The  short  period  is  |.    The  short  ratio  (4%) 

will  be  1.0166^.    The  short  interest  (5%)  will  be  .02083^. 

We  first  ascertain  the  value  for 

the  ten  full  periods,  viz.,  for  $1. .   1.0449129 

Add  to  this  the  short  interest.  . .      .0208333 

1.0657462 

and  divide  by  the  short  ratio 1.0166667 

To  perform  this  division  it  will  be  easier  to  divide  3  times  the 
dividend  by  3  times  the  divisor. 

3.05  )  3.1972386  (  Quotient  1.0482750 
3.05 

1472 
1220 

2523 
2440 

838 
610 

2286 
2135 

151 
152 

Multiplying  down  by  the  usual  procedure,  we  have  the  follow- 
ing schedule:  /j\ 

SHORT  TKRMINAI,. 


Total 

Net 

■D^^Ay 

Date 

Interest 
5% 

come 
4% 

Amornsation       ^^^^^                   Par 

1904  May  1 

104,827.50     100,000.00 

Nov.  1 

2,500.00 

2,096.55 

403.45 

104,424.05 

1905   May  1 

2,500.00 

2,088.48 

411.52 

104,012.53 

Nov.  1 

2,500.00 

2,080.25 

419.75 

103,592.78 

1906   May  1 

2,500.00 

2,071.86 

428.14 

103,164.64 

Nov.  1 

2,500.00 

2,063.29 

436.71 

102,727.93 

1907   May  1 

2,500.00 

2,054.56 

445.44 

102,282.49 

Nov.  1 

2,500.00 

2,045.65 

454.35 

101,828.14 

1908   May  1 

2,500.00 

2,036.56 

463.44 

101,364.70 

Nov.  1 

2,500.00 

2,027.29 

472.71 

100,891.99 

1909   May  1 

2,500.00 

2,017.84 

482.16 

100,409.83 

Oct.   1 

2,083.33 

1,673.50 

409.83 

100,000.00 

27,083.33 

22,255.83 

4,827.50 

48  Thk  Accountancy  ov  Invkstmknt. 

113. — Rule  for  Short  Terminals.  Ascertain  the  value 
for  the  full  periods,  disregarding  the  terminal.  To  this  value 
add  the  short  interest  and  divide  by  the  short  ratio. 

114. — It  may  be  remarked  that  the  same  process  applies  to 
short  initial  periods,  or  even  to  bonds  originally  issued  between 
interest  dates,  and  also  maturing  between  interest  dates.  In 
the  latter  case  it  would  be  applied  twice. 

115.— Discounting.  Hitherto  we  have  calculated  the 
longest  period,  and  then  obtained  the  shorter  ones  by  multipli- 
cation and  subtraction.  We  can  also  work  backwards,  obtain- 
ing each  value  from  the  next  later  by  addition  and  division. 

Thus,  beginning  at  maturity  with  par 100,000.00 

and  adding  to  it  the  coupon  then  due 2,500 .  00 

102,500.00 
We  discount  this  by  dividing  by  1.02,  which  gives  100,490.20 
the  value  one  period  before  maturity.     To  obtain 
the  next  value  add  the  coupon 2,500 .  00 

102,990.20 
Divide  again  by  1 .  02,  giving  the  next  previous  value  100,970 .  78 
Thus  successive  terms  may  be  obtained  as  far  as  desired. 

116. — In  the  last  half  year  of  a  bond,  its  value  should  be 
discounted,  and  not  found  as  in  Article  97.  Thus,  if  the 
bond  in  question  were  sold,  when  only  three  months  remained 
to  maturity,  $102,500  would  be  divided  by  101,  which  would 
give  $101, 485. 15  "flat,"  equivalent  to  $100, 235. 15  and  interest; 
whereas,  by  the  ordinary  rule  it  would  be  $100,245.10.  The 
theoretically  exact  value  (recognizing  effective  rates,  which  is 
never  done  in  business)  would  be  $100,240.12.  To  ''split  the 
difEerence  "  would  be  an  easy  way  of  adjusting  the  matter  and 
would  be  almost  exact. 

117. — Serial  Bonds  and  Various  Maturities.  Bonds 
are  often  issued  in  series.  For  example:  $30,000,  of  which 
$1,000  is  payable  after  one  year,  another  $1,000  after  two  years, 
and  the  last  $1,000  30  years  from  date.  Other  series  are  more 
complex;  as,  $2,000  each  year  for  5  years,  and  $4,000  each  year 
thereafter  for  5  years.  After  finding  the  initial  value  for  each 
instalment  and  adding  these  together,  the  aggregate  may,  for 
purposes  of  deriving  successive  values,  be  treated  as  a  unit,  and 
multiplied  down  in   one   process.     The  principles   in   Article 


Valuation  of  Bonds.  49 

103  must  be  observed  as  to  that  part  of  the  par  value,  if  any, 
which  comes  due  between  balancing  periods. 

118. — Investment  of  Trust  Funds.  A  bond  which  has 
been  purchased  by  a  trustee  at  a  premium  is  subject  to  amorti- 
sation in  the  absence  of  testamentary  instructions  to  the  con- 
trary.    (Hardcastle  on  Accounts  of  Executors,  p.  49.) 

The  trustee  has  no  right  to  pay  over  the  full  cash  interest, 
because  he  must  keep  the  principal  intact  for  the  remainder 
man.  If,  for  example,  he  were  to  invest  $104,491.29  in  a  5% 
bond  having  five  years  to  run,  and  the  life  tenant  were  to  die  at 
the  end  of  five  years,  the  fund  would  be  depleted  by  $4,491.29, 
to  the  injury  of  the  remainder  man.  Since  this  is  a  4%  basis, 
he  should  pay  over  at  the  end  of  the  first  half  year  only  4%  of 
$104,491.29  (=$2,089.83),  not  5%  of  $100,000  (=$2,500). 
He  then  has  $410 .  17  cash  to  re-invest,  and  the  fund,  including 
this,  is  still  $104,491.29.  It  may  be  difficult  to  invest  the 
$410 .  17  at  as  favorable  a  rate  as  the  bonds,  very  small  and  very 
large  amounts  being  most  difficult  to  invest.  He  can,  at  least, 
deposit  it  in  a  trust  company  and  receive  interest  at  some  rate  or 
other. 

119. — At  the  end  of  the  second  half-year  the  bond  interest 
is  only  $2,081.62;  but  the  beneficiary  is  entitled,  also,  to  the 
interest  on  the  $410 .  17.  If  this  has  been  re-invested  at  exactly 
4%,  the  interest  thereon  is  $8 .  20,  and  the  total  payable  to  the 
beneficiary  is  $2,081.62  +  8.20  =  $2,089.82,  practically  the 
same  as  before,  and  $418.38  is  deposited  or  invested  as  before. 
He  now  has  in  the  fund  $103,662.74  +  410.17  +  418.38  = 
104,491 .  29.  He  has  paid  over  all  the  new  interest  earned,  and 
he  has  kept  the  corpus  or  principal  intact. 

Suppose,  however,  he  was  not  able  to  get  4%  for  the 
$410.17,  but  only  3%,  so  that  from  this  source  would  come 
only  $6.15,  making  the  total  income  $2,081.62  +  6.15  = 
$2,087.77.  There  is  a  slight  falling  off  in  income,  but  that  is 
to  be  expected  when  part  of  an  investment  is  returned  and  re- 
invested at  a  lower  rate.  If  the  re-investment  had  been  at 
4>^%,  the  income  would  have  been  $2,090.74,  slightly  more 
than  the  first  half  year,  owing  to  the  improved  demand  for 
capital.  It  might  be  urged  that  the  beneficiary  ought  to  receive 
$2,089.83  periodically  —  no    more,  no    less  —  being   4%    on 


50  The  Accountancy  of  Investment. 

$104,491.29.  This  would  leave  $410.17  each  half-year  to  be 
invested  in  a  sinking  fund,  from  which  no  interest  should  be 
drawn,  but  which  should  be  left  to  accumulate  to  maturity, 
when  it  would  exactly  replace  the  premium,  if  compounded  at 
4%.  But  this  hope  might  not  be  realized.  Very  likely  the 
average  rate  would  be  less  or  more  than  4%.  If  less,  the  original 
fund  would  be  to  some  extent  depleted,  and  the  remainder  man 
wronged;  if  more,  there  is  too  much  in  the  fund,  and  the  life 
tenant  has  received  too  little.  It  seems,  therefore,  that  the 
sinking  fund  principle  is  not  correct  in  a  case  like  this,  and  that 
we  should  rather  recognize  a  gradual  disappearance  of  capital 
than  constitute  a  fictitious  sinking  fund. 

120. — Prof.  Hardcastle's  example    (p.    50)    expanded  to 
$10,000  instead  of  $100  — would  be  scheduled  thus: 


Coupon. 

Income. 

Cash. 

Bond. 

200.00 
200.00 
200.00 
200.00 

1800.00 

152.94 
152.13 
151.47 
150.74 

607.28 

47.06 
47.87 
48.53 
49.26 

192.72 

10.192.72 
10,145.66 
10,097.79 
10,049.26 
10,000.00 

The  life  tenant  would  receive,  at  the  end  of  the  first  half- 
year,  $152.94;  at  the  end  of  the  second,  $152.13  +  whatever 
the  $47 .  06  cash  had  earned ;  at  the  end  of  the  third,  $151 .  47  ■+- 
whatever  $94.93  had  earned;  at  the  close,  $150.74  +  whatever 
$152.46  had  earned;  and  if  the  cash  balance  was  constantly 
deposited  in  the  trust  company  at  3%,  the  life  tenant  would 
receive  a  uniform  income  of  $152 .  94. 

121. — In  a  case  reported  in  the  State  of  New  York  (38 
App.  Div.  419),  Justice  CuUen  very  clearly  lays  down  the  law 
as  to  the  duty  of  the  trustee  to  reserve  a  part  of  the  interest 
to  provide  for  the  premium,  and  says  that  **  any  other  view 
would  lead  to  the  impairment  of  the  principal  of  the  trust,  to  pro- 
tect the  integrity  of  which  has  always  been  the  cardinal  rule  of 
courts  of  equity."  He  says  further:  *'If  one  buys  a  ten- year 
five  per  cent,  bond  at  one  hundred  and  twenty,  the  true  income 
or  interest  the  bond  pays  is  not  4^^%  on  the  amount  invested, 
nor  5%  on  the  face  of  the  bond,  but  2^\%  on  the  investment, 
or  d^^^Q%  on  the  face  of  the  bond.     The  matter  is  simply  one 


Vai^uation  of  Bonds.  51 

of  arithmetical  calculation,  and  tables  are  readily  accessible, 
showing  the  result  of  the  computation. ' ' 

122. — The  learned  judge's  example  would  work  out  in  a 
schedule  as  follows,  with  a  slight  correction  in  the  initial 
figures,  and  applying  it  to  a  par  of  $100,000: 


Total 
Interest 

Income 
Paid  Over 

Re-invested. 

Present 
Value 

120,039.00    ' 

2,500.00 

1,620.52 

879.48 

119,159.52   ' 

2.500.00 

1,608.66 

891.34 

118,268.18 

2,500.00 

1,596.62 

903.38 

117,364.80 

2,500.00 

1,584.42 

915.58 

116,449.22 

2,500.00 

1,572.07 

927.93 

115,521.29 

2,500.00 

1,559.53 

940.47 

114,580.82 

2,500.00 

1.546.85 

953.15 

113,627.67 

2,500.00 

1,533.97 

966.03 

112,661.64 

2,500.00 

1,520.93 

979.07 

111,682.57 

2,500.00 

1,507.72 

992.28 

110,690.29 

2,500.00 

1,494.31 

1,005.69 

109,684.60 

2,500.00 

1,480.75 

1,019.25 

108,665.35 

2,500.00 

1,466.93 

1,033.07 

107,632.28 

2,500.00 

1,453.08 

1,046.92 

106,585.36 

2,500.00 

1,438  91 

1,061.09 

105,524.27 

2,500.00 

1,424.57 

1,075.43 

104,448.84 

2,500.00 

1,410.06 

1,089.94 

103,358.90 

2,500.00 

1,395.35 

1,104.65 

102,254.25 

2,500.00 

1,380.43 

1,119.57 

101,134.68 

2,500.00 

1.365.32 

1,134.68 

100,000.00 

50,000.00 

29,961.00 

20,039.00 

123. — This  is  perfectly  correct,  but  we  can  scarcely  agree 
with  the  method  described  further  on  in  the  same  opinion,  as 
follows:  **  There  is,  however,  a  simpler  way  of  preserving  the 
principal  intact  —  the  method  adopted  by  the  learned  referee. 
He  divided  the  premium  paid  for  the  bonds  by  the  number  of 
interest  payments,  which  would  be  made  up  to  the  maturity  of 
the  bonds,  and  held  that  the  quotient  should  be  deducted  from 
each  interest  payment  and  held  as  principal.  These  deductions 
being  principal,  the  life  tenant  would  get  the  benefit  of  any 
interest  that  they  might  earn.  We  do  not  see  why  this  plan 
does  not  work  equal  j  ustice  between  the  parties. '  *  The  reason 
**why  it  does  not  work  equal  justice"  is  that  the  life  tenant 
in  the  earlier  years  receives  much  less  than  his  due  share  of 
the  income,  but  from  year  to  year  he  gradually  receives  more 
and  more,  until  he  receives  more  than  his  share,  but  not  till  the 
very  last  payment  has  he  overtaken  his  true  share.  Thus,  if 
he  dies  before  the  maturity  of  the  bonds,  it  is  certain   that 


52  Thb;  Accountancy  of  Investment. 

"equal  justice"  will  not  have  been  done,  but  the  remainder 
man  would  have  altogether  the  best  of  it. 

124. — To  particularize,  the    ''referee's   plan"   would   be 
scheduled  as  follows: 

Trust  Fund.     "Referee's  Plan." 


Total 
Interest 

Income 
Paid  Over 

Re-invested 

Present 
Value 

120,039.00 

2,500.00 

1,498.05 

1,001.95 

119,037.05 

2,500.00 

1,498.05 

1,001.95 

118,035.10 

2,500.00 

1,498.05 

1,001.95 

117,033.15 

2,500.00 

1,498.05 

1,001.95 

116,031.20 

2,500.00 

1,498.05 

1,001.95 

115,029.25 

2,500.00 

1,498.05 

1,001.95 

114,027.30 

2,500.00 

1,498.05 

1,001.95 

113,025.35 

2,500.00 

1,498.05 

1,001.95 

112,023.40 

2,500.00 

1,498.05 

1,001.95 

111,021.45 

2,500.00 

1,498.05 

1,001.95 

110,019.45 

&c. 

&c. 

&c. 

&c. 

It  is  unnecessary  to  continue  this  further,  but  by  com- 
paring it  with  our  schedule  (K)  it  will  be  seen  that,  if  the 
remainder  man  received  the  fund  after  five  years,  it  would  be 
at  such  a  valuation  on  the  bonds  that  he  could  enjoy  an  income 
of  over  2 .  80  per  cent,  for  the  other  five  years,  and  yet  keep  the 
principal  intact.  The  method  of  the  referee  is  false  and 
arbitrary. 

125. — Single  Column  Schedule.     Instead  of  distributing 

the  figures  of  the  schedule  into  four  columns,  as  in  our  examples, 

it  is  frequently  easier  to  ignore  the  amortisation  column  and 

simply  add  the  net  income,  then  subtract  the  cash  interest. 

Thus,  Schedule  A  would  begin  as  follows: 

104,491.29 
plus  2,089.83 

106,581.12 
minus         2,500.00 

104,081.12 
plus  2,081.62 

106,162.74 
minus         2,500.00 

103,662.74 
etc. 

The  Book  Values  may  then  be  set  down  at  once;  the  amorti- 
sations will  be  their  differences;  and  the  Net  Income  will  be  the 
Cash  Interest  minus  the  Net  Income.     By  using  red  ink  for 


Vai^uation  op  Bonds.  53 

the  subtrahends  (which  we  indicate  by  italic  figures)  setting 
them  down  in  advance  on  the  proper  lines,  the  addition  and 
subtraction  can  be  performed  at  one  operation.     For  example: 

104,491.29 

2,089.83 
2,500.00 


104,081 

.12 

2,081. 

62 

2.500  00 

103,662, 

.74 

2,073.35 

2,500 

00 

103,236.09 
etc. 

It  will  be  noticed  that  the  computation  of  the  interest  is  done 
without  using  any  other  paper.  Even  with  a  fractional  rate, 
like  that  in  Judge  CuUen's  example,  2.7%'  per  annum,  or  1 .  35% 
per  period,  the  1%,  the  .3%  and  the  .05%  can  be  successively 
written  down  direct: 

120,039.00 
1,200.39 
360.117 
60.019 
2.500  00 


119,159.526 

1,191.595 

357.479 

59.580 

2,500.00 

118,268.180 

etc. 

126. — Irredeemable  Bonds.  Sometimes,  as  in  the  British 
Consols,  there  is  no  right  nor  obligation  of  redemption.  If 
the  government  wishes  to  pay  off  it  has  to  buy  at  the  market 
price.  There  is,  then,  no  question  of  amortisation;  the  invest- 
ment is  simply  a  perpetual  annuity.  The  cash  interest  is  all 
revenue,  and  the  original  cost  is  the  constant  book  value.  If 
^100  of  4%  consols  be  bought  at  96,  the  income  is  ^4  per 
annum;  the  book  value  is  ^96.  As  an  investment  of  ^"96 
produces  /^  4,  the  rate  of  income  is  4  -f-  96  =  .  04 J . 

127. — Optional  Redemption.  Sometimes  the  issuer  has 
the  rtg-ht  to  redeem  at  a  certain  date  earlier  than  the  date  at 


54  I^HK  Accountancy  of  Investment. 

which  he  must  redeem.  It  must  always  be  expected  that  this 
right  will  be  exercised  if  profitable  to  the  issuer;  hence,  bonds 
bought  at  a  premium  must  be  considered  as  maturing,  or 
reaching  par,  at  the  earlier  date.  Bonds  bought  at  a  discount 
must  be  considered  as  running  to  the  longer  date. 

The  option  of  redemption  is  sometimes  attended  by  a 
premium.  For  example:  the  issuer  of  a  thirty-year  bond  re- 
serves the  right  to  redeem  after  twenty  years  at  105.  Unless 
you  buy  at  such  a  basis  that  after  ten  years  the  book  value  will 
be  105  or  more,  this  redemption  right  is  a  detriment.  You 
must,  in  that  case,  consider  that  you  are  buying  a  twenty-year 
bond,  and  that  the  par  is  1 .  05  times  the  nominal  par. 

There  is  also  a  form  of  bond-issue,  not  uncommon  in  Europe, 
where  a  certain  or  indefinite  number  of  bonds  is  drawn  by  lot 
each  year  to  be  paid  off.  As  they  are  usually  issued  at  a  dis- 
count, the  earlier  drawn  bonds  are  the  more  profitable.  The 
investor,  in  estimating  his  income,  must  assume  that  his  bonds 
will  be  the  last  ones  drawn.  If  drawn  earlier,  there  is  a  profit 
exactly  the  same  as  that  arising  from  a  sale  above  book  value. 

Note. — When  quarterly  bonds  are  offered  in  competition  with  those 
on  which  interest  is  paid  half-yearly,  it  is  desirable  to  know  how  much  is 
added  to  the  value  by  the  fact  of  quarterly  conversion.  For  this  purpose 
Appendix  2  gives  a  series  of  multipliers  —  decimal  fractions  by  which  the 
premium  or  discount  is  multiplied  to  give  the  increment  thus  added  to 
the  value  of  a  semi-annual  bond.  At  the  resulting  price  the  quarterly 
bond  will  pay  an  effective  income  equivalent  to  that  of  the  semi-annual 
bond.  Thus,  a  3>^%  bond  35  years,  yielding  2>^%  income,  payable  semi- 
annually, is  worth  1 .  23234838;  the  premium  is  .  23234838.  In  Appendix  2, 
on  the  line  2.50,  in  the  column  3^,  we  find  the  multiplier  .0109035,  by 
which  we  multiply  .23234838,  giving  .00253341  as  the  increment.  This, 
added  to  1.23234838,  makes  1.23488179  as  the  value  of  a  quarterly  bond, 
which  will  be  exactly  as  profitable  as  the  semi-annual  bond  at  1 .  23234838. 
If  the  same  bond  were  to  yield  4^%,  we  have  the  value  for  semi-annual 
interest  .82458958,  or  a  discount  of  .17541042,  which  is  multiplied  by  the 
decimal  .019578,  giving  the  increment  .00343419,  and  the  value  of  the 
quarterly  bond,  .82802377  {=  .82458958+  .00343419). 

Appendix  3  gives  in  condensed  and  progressive  form  the  processes 
explained  in  the  previous  chapters. 


'    B  R  Af?  ,-^^ 
OF  THE 

OF 

Thk  Accountancy  of  iNVESTfelfclSSfiS-- 


CHAPTER  IX. 

Forms  of  Account  —  Generai,  Principles. 

128. — In  any  system  of  accountancy  on  an  extensive  scale, 
in  order  to  fulfill  the  opposite  requirements  of  minuteness  and 
comprehensiveness,  it  is  necessary  to  keep,  in  some  form,  a 
General  Ledger  and  various  Subordinate  Ledgers.  Each 
account  in  the  General  Ledger,  as  a  rule,  comprises  or  sum- 
marizes the  entire  contents  of  one  Subordinate  Ledger.  The 
General  Ledger  accounts  deal  with  whole  classes  of  like  nature; 
the  Subordinate  Ledger  with  each  individual  asset  or  liability, 
or  with  groups  which  may  be  treated  as  individuals.  It  is  the 
province  of  the  General  Ledger  to  give  information  in  grand 
totals  as  an  indicator  of  tendencies;  it  is  the  function  of  the 
Subordinate  Ledger  to  give  every  desired  information  as  to 
details  even  beyond  the  figures  required  for  balancing  —  facts 
not  of  numerical  accountancy  only,  but  descriptive,  cautionary, 
auxiliary.  Thus  the  General  Ledger  may  contain  an  account, 
**  Mortgages,"  which  will  show  the  increase  and  decrease  of  the 
amount  invested  on  mortgage,  and  the  resultant  or  present 
amount ;  the  Mortgage  Ledger  will  contain  an  account  for  each 
separate  mortgage,  with  additional  information  as  to  interest, 
taxes,  insurance,  title,  ownership,  security,  valuation,  and  any 
thing  useful  or  necessary  to  be  known. 

129. — We  shall  assume  that  a  General  Ledger  exists  with 
Subordinate  or  Class  Ledgers.  We  shall  also  assume  that  the 
accounts  are  to  be  so  arranged  as  to  give  currently  the  amount 
of  interest  earned  and  accruing,  the  amount  which  has  accrued 
up  to  any  time,  and  the  amount  outstanding  and  overdue  at  any 
time.  It  hardly  seems  necessary  to  argue  this  point,  were  it 
not  that  many  important  investors  pay  no  attention  to  interest 
until  it  matures,  and  some  do  not  carry  it  into  account  until  it 
is  paid.  They  are  compelled  to  make  an  adjustment  on  their 
periodical  balancing  dates  "in  the  air,"  compiling  it  from 
various  sources  without  check,  which  seems  as  crude  as  it  would 
be  to  take  no  account  of  cash,  except  by  counting  it  occasionally. 


56  The  Accountancy  of  Investment. 

The  Profit  and  lyoss  account  depends  for  its  accuracy  upon  the 
interest  earned,  not  upon  the  interest  falling  due,  nor  upon  the 
interest  collected,  and  the  accruing  of  interest  is  a  fact  which 
should  be  recognized  and  recorded. 

130. — In  considering  the  forms  of  account  for  investments, 
we  will  first  take  up,  as  being  simpler,  those  in  which  there  is 
never  any  value  to  be  considered  other  than  par,  such  as  direct 
mortgages  and  loans  upon  collateral  security.  Both  of  these 
classes  of  investment  are  for  comparatively  short  terms,  and  are 
usually  the  result  of  direct  negotiation  between  borrower  and 
lender,  not  the  subject  of  purchase  and  sale;  hence,  changes  in 
rate  of  interest  are  readily  effected  by  agreement,  and  do  not 
result  in  a  premium  or  discount. 


Thk  Accountancy  of  Investment.  57 


CHAPTER  X. 

Rkai.  Estate  Mortgages. 

131. — The  instruments  which  we  have  spoken  of  as  '  'bonds' ' 
are  very  often  secured  by  a  mortgage  of  property.  But  one 
mortgage  will  secure  a  great  number  of  bonds,  the  mortgagee 
being  a  trustee  for  all  the  bondholders.  The  instruments  of  which 
we  now  speak  are  the  ordinary  "bond  and  mortgage,"  by 
which  the  investor  receives  from  the  borrower  two  instruments: 
the  one  an  agreement  to  pay,  and  the  other  conferring  the  right, 
in  case  default  is  made,  to  have  certain  real  estate  sold,  and  the 
proceeds  used  to  pay  the  debt.  As  only  a  portion  of  the  value 
of  the  real  estate  is  loaned,  the  reliance  is  primarily  on  the  mort- 
gage rather  than  on  the  bond.  Therefore,  the  mortgagee  must 
be  vigilant  in  seeing  that  his  margin  is  not  reduced  to  a  haz- 
ardous point.  This  may  happen  by  the  depreciation  of  the 
land  for  various  economic  reasons;  by  the  deterioration  of  the 
structures  thereon,  through  time  or  neglect;  by  destruction 
through  fire  or  by  the  non-payment  of  taxes,  which  are  a  lien 
superior  to  all  mortgages.  By  reason  of  these  risks  a  mortgage 
loan  is  seldom  made  for  more  than  a  few  years;  but  after  the 
date  of  maturity,  extensions  are  made  from  time  to  time;  or, 
even  more  frequently,  without  formal  extension,  it  is  allowed 
to  remain  ''on  demand,"  either  party  having  the  right  to  ter- 
minate the  relation  at  will.  A  large  proportion  of  outstanding 
mortgages  are  thus  "on  sufferance,"  or  payable  on  demand. 
The  market  rate  of  interest  seldom  causes  the  obligation  to 
change  hands  at  either  a  premium  or  a  discount,  hence  we 
may  ignore  that  feature,  referring  the  exceptional  cases,  where 
it  occurs,  to  the  analogy  of  bonds. 

The  two  instruments,  bond  and  mortgage,  relate  to  the 
same  transaction,  are  held  by  the  same  owner,  and  for  most 
purposes  are  treated  as  a  unit.  In  bookkeeping,  the  investment 
must  likewise  be  treated  as  a  unit,  both  as  to  principal  and 
income. 

132. — It  is  desirable  to  know  at  any  time  how  much  is  due 
on  principal,  allowing  for  any  partial  payments.     It  is  also  de- 


58  The  Accountancy  of  Investment. 

sirable  to  know  what  interest,  if  any,  is  due  and  payable,  and 
to  be  able  to  look  to  its  collection.  An  account  of  principal 
and  an  account  of  interest  are,  therefore,  requisite.  It  is  better, 
however,  that  these  two  accounts  for  the  same  mortgage  be 
adjacent. 

133. — Accrued  interest  need  not  be  considered  as  to  each 
mortgage.  It  should  be  treated  in  bulk,  as  the  revenue  of  the 
aggregate  mortgages,  as  will  be  explained  hereafter.  The  In- 
terest account  here  referred  to  is  debited  on  the  day  when  the 
interest  becomes  a  matured  obligation,  and  credited  when  that 
obligation  is  discharged. 

134. — Those  who  adhere  to  the  original  form  of  the  Italian 
ledger  will  probably  be  averse  to  combining  with  the  ledger- 
account  any  general  business  information;  in  fact,  that  form  is 
not  suited  for  such  purposes,  and  is  not  adapted  to  containing 
anything  but  the  bare  figures  that  will  make  the  trial-balance 
prove.  But  the  modern  conception  of  a  ledger  is  broader  and 
more  practical:  it  should  be  an  encyclopedia  of  information 
bearing  on  the  subject  of  the  account;  it  should  be  specialized 
for  every  class-ledger;  it  should  be  of  any  form  which  will  best 
serve  its  purposes,  regardless  of  custom  or  tradition. 

135. — The  form  of  mortgage-ledger  which  seems  to  the 
author  to  be  the  best,  contains  four  parts :  1.  Descriptive. 
2.  Account  of  principal.  3.  Account  of  interest.  4.  Aux- 
iliary information.  These  may  occupy  four  successive  pages,  or 
two  pages,  if  preferred.  In  the  latter  case,  if  kept  in  a  bound 
volume,  the  arrangement  whereby  two  of  these  parts  should  be 
on  the  left-hand  page  and  two  on  the  right,  confronting  each 
other,  is  a  convenient  one,  giving  all  the  facts  at  one  view.  For 
a  loose-leaf  ledger  the  order  1,  2,  3,  4  will  generally  be  found 
the  best. 

136. — Mortgages  should  be  numbered  in  chronological 
order,  and  every  page  or  document  should  bear  the  number  of 
the  mortgage  loan  to  which  it  refers. 

137. — The  account  of  principal  (Part  2)  may  be  in  the  ordi- 
nary ledger  form;  but  what  is  known  as  the  balance  column,  or 
three  column  form,  will  be  found  more  convenient.  It  contains 
but  one  date  column,  so  that  successive  transactions,  whether 


Real  Estate  Mortgages.  59 

payments  on  account,  or  additional  sums  loaned,  appear  in  their 
proper  chronological  order. 

138. — The  mortgage  usually  contains  clauses  which  permit 
the  mortgagee,  when  the  mortgagor  fails  to  make  any  payment 
for  the  benefit  of  the  property,  like  taxes  and  insurance 
premiums,  to  step  in  and  advance  the  money,  which  he  has  the 
right  to  recover  with  interest.  It  will  be  useful  to  have  columns, 
also,  for  these  disbursements  and  their  reimbursements.  The 
complete  Part  2  will  then  take  the  form  shown  on  page  62. 

139. — The  Interest  account  (Part  3)  may  be  very  simple. 
It  contains  two  columns,  one  for  debits  on  the  day  when  interest 
falls  due,  the  other  for  crediting  when  it  is  collected.  The 
entries  in  the  Interest  account  will  naturally  be  much  more 
numerous  than  in  the  Principal  account;  hence,  this  pair  of 
columns  may  be  repeated  several  times.  The  arrangement 
shown  on  page  63  has  been  found  advantageous. 

140. — Experience  shows  that  the  safest  way  to  ensure 
attention  to  the  punctual  and  accurate  collection  of  interest  is 
to  charge  up,  systematically,  under  the  due  date,  every  item,  and 
let  it  stand  as  a  debit  balance  until  collected.  Many  attempt 
to  accomplish  the  same  purpose  by  merely  marking  **paid" 
on  a  list,  but  this  is  apt  to  lead  to  confusion,  and  it  is  difficult 
afterward  to  verify  the  state  of  the  accounts  on  any  given  date. 

141. — It  is  not  proposed  in  this  treatise  to  prescribe  the 
forms  of  Posting  Medium  (Cash  Book,  Journal,  etc.)  from  which 
the  postings  in  the  ledger  are  made,  because  these  forms  are, 
in  recent  times,  so  largely  dependent  upon  the  peculiarities  of 
the  business,  and  have  deviated  so  far  from  the  traditional 
Italian  form,  that  no  universal  type  could  be  presented.  We 
shall,  however,  give  the  debit  and  credit  formulas  underlying 
the  postings,  and  will  suggest  auxiliary  books  or  lists  for  making 
up  the  entries. 

142.— The  formula  for  the  '*  Due  "  column  of  the  Interest 
account  is: 

Interest  Due  /  Interest  Accrued  $ 

It  is  a  transfer  from  one  branch  of  Interest  Receivable,  viz. , 
that  which  is  a  debt,  but  not  yet  enforceable,  to  another  branch, 
viz.,  that  which  is  a  matured  claim. 


6o 


Tnn  Accountancy  of  Investment. 


143. — In  the  General  I^edger  the  entry  will  be  simply  as 
above: 

Interest  Due  /  Interest  Accrued  $... 

and  this  may  be  a  daily,  a  weekly,  a  monthly  entry,  or  for  any 
other  space  of  time,  according  to  the  general  practice  of  the 
business;  the  monthly  period  is  most  in  use,  and  we  shall  take 
that  as  the  standard.  The  credit  side  of  the  entry  (/  Interest 
Accrued)  is  not  regarded  in  the  Subordinate  I^edger  (Article 
130),  but  the  debit  entry  (Interest  Due  /  )  must  be  somewhere 
analysed  into  its  component  parts;  in  other  words,  there  must 
be  somewhere  a  list,  the  total  of  which  is  the  aggregate  falling 
due  on  all  mortgages,  and  the  items  of  which  are  the  interest 
falling  due  on  each  mortgage. 

144. — The  following  headings  will  suggest  the  require- 
ments for  such  a  list,  the  form  to  be  modified  to  conform  to  the 
general  system. 

REGISTER  OF  INTEREST  DUE. 

MORTGAGES. 


Date 


No. 


Principal 


Rate 


Time 


Interest 


Total 


145. — Part  1  of  the  Mortgage  account  is  descriptive.  Its 
elements  may  be  placed  in  various  orders  of  arrangement.  It 
is  believed  that  the  form  on  page  61  combines  all  the  particulars 
ordinarily  required  in  the  State  of  New  York. 

146. — Part  4  is  not  an  essential  feature,  and  may  be  re- 
placed by  card-lists,  if  preferred.  Yet,  if  there  be  space,  there 
are  advantages  in  having  all  the  information  about  a  certain 
mortgage  accessible  at  one  time,  and  concentrated  in  one  place. 
The  changing  names  and  addresses  of  the  mortgagors  and 
owners,  and  the  successive  policies  of  insurance  require  for 
their  record  considerable  space,  which  may  be  arranged  under 
the  headings  on  page  64. 

147. — The  card-form  of  Mortgage  Ledger  is  also  very  con- 
venient in  many  respects,  and  the  forms  here  given  may  be  re- 
arranged so  as  to  suit  different  sizes  of  cards.     Both  in  cards 


Rkai.  Estat:^  Mortgages. 


6i 


V. 

^ 

a> 

C<3 

t 

Ph 

^    « 

-> 

M    ^ 

WP 

d 
.^'^ 

o     ^ 

^  ^  a 


o 

>^ 

'o 


O    N    00    OS    O 

iH     iH     rH     rH     CM 

03    Q    O    0)    G) 


rH    CI     W    rjl    to 

rl     iH     i-H     r-l     iH 
03     0)     OS     0>     O) 


©  N  00  OS  O 

o  O  O  O  >H 

0)  OS  OS  OS  OS 

iH  rH  fH  r-t  tH 


i-i  01  CO  t|<  lO 
O  O  O  O  O 
O    A    OS    OS    OS 


/OF  THE  \ 


62 


Thk  Accountancy  of  Investment. 


0 

i 

0 

> 

3 

M 

> 

s 

> 
a 

n 

w 
a 

0   pi 

o   > 

d   !^ 
w  o 

g  ^^ 

2! 

H 

0 

Rkai.  Estatb  Mortgages. 


63 


64 


The  Accountancy  op  Investment. 


0 

> 

> 

I 

i 

o 
pi 

W 

§ 

d 

0 
1— I 

i 

o 
o 

1 

- 

> 

0 
0 

w 

03 

o 

o 

t— ( 
o 

1 

o 
p 

> 

o 
a 

!^ 

o 

0 

8 

o 

> 


Rkai.  Estate  Mortgages.  65 

and  loose  leaves  it  will  be  helpful  to  use  different  colors  for 
pages  of  different  contents.  Where  interest  on  different  mort- 
gages falls  due  in  different  months,  tags  marked  "J  J,"  "FA," 
"MS,"  "AO,"  "MN,"  "  J  D,"  may  project  from  the  interest- 
sheet  like  an  index,  the  tags  of  each  month  at  the  same  dis- 
tance from  the  top.  This  will  greatly  facilitate  the  compiling 
of  the  Register  of  Interest  Due. 

148. — The  Interest  Register  should  always  be  made  up  and 
proved  (subject  to  modifications)  in  advance.  In  doing  this, 
instead  of  making  the  computations  in  the  register  and  posting 
thence  to  the  ledger,  a  surer  way  is  by  "reverse posting";  that 
is,  making  the  computation  from  the  data  in  the  ledger  and 
entering  it  there  at  once,  in  pencil  if  preferred;  then  copying 
the  items  into  the  register,  where  the  total  can  be  proved. 
When  this  has  been  done,  we  can  be  sure  without  further  check 
that  the  ledger  is  correct. 

149. — It  is  desirable,  also,  to  have  receipts  prepared  in  ad- 
vance ready  for  signature.  The  correctness  of  these  receipts 
may  be  assured  by  introducing  them  into  the  ' '  reverse  posting ' ' 
process,  as  follows :  Having  made  the  computation  on  the 
ledger,  prepare  the  receipts /^-^m  the  ledger,  copying  down  the 
figures  just  as  they  appear;  from  the  receipts  make  up  the 
register,  which  prove  as  before.  This  method  may  be  extended 
to  the  notices,  if  any  are  sent  to  the  mortgagors,  the  notice 
being  derived  from  I^edger  account,  the  receipt  from  the  notice, 
and  the  register  from  the  receipt;  if  the  register  proves  correct, 
the  correctness  of  its  antecedents  is  established.  These  interest 
notices  may  be  made  of  assistance  in  the  book-keeping,  if  their 
return  is  insisted  upon  and  made  convenient.  Below  the  formal 
notification  of  the  sum  falling  due  on  such  a  date,  with  all  par- 
ticulars, is  a  blank  form  something  as  follows  : 

*  'In  payment  of  the  above  interest  I  inclose  check  on  the 

...for    $ and 

request  you  to  acknowledge  receipt  as  below. 

[Signature) 

Address " 


66 


Th^  Accountancy  of  Investment. 


The  notice  upon  its  being  received,  together  with  the  check, 
becomes  a  **  voucher- with-cash,"  and  the  credits  on  the  cash 
book,  and  the  interest  page  of  the  Mortgage  I^edger  are  made 
directly  from  the  documents.  Book-to-book  posting,  which 
formerly  was  the  only  method  of  re-arranging  items,  is  becoming 
obsolete,  being  superseded  in  many  businesses  by  voucher  or 
document  posting.  By  the  carbon  process  the  notice  and  the 
receipt  may  be  filled  in  simultaneously  in  fac- simile. 

150. — General  Ledger,  Mortgages  Account.  The  Class 
account  * '  Mortgages ' '  in  the  General  I^edger  is  simply  kept  to 
show  aggregates.  Its  entries  are,  as  far  as  possible,  monthly, 
the  posting-mediums  being  so  arranged  as  to  give  a  monthly 
total  of  the  same  items  which  have  already  been  posted  in 
detail  to  the  Mortgage  Ledger.  The  standard  form  of  Ledger 
account  may  be  used,  or  the  three  column.  In  the  former,  the 
debits  and  credits  of  the  same  month  should  be  kept  in  line, 
even  though  one  line  of  paper  be  wasted. 


Dr. 


[Form  1.] 
MORTGAGES. 


Cr. 


1904 
Jan. 

Feb. 

March 

April 

May 

June 


July 


0 
1-31 
1-29 
1-31 
1-30 
1-31 
1-30 


Balance 
Total  loaned 


Balance 


169.000 
12,000 
10,000 
50.000 
20.000 
5,000 
10.000 


276,000 
182.000 


00 


Jan. 

March 
April 
May 
June 


1-31 

1  31 
1-30 
1-31 
1-30 
30 


Total  paid  in 
Balance 

7,000 

32,000 

40,000 

12  0  0 

3  000 

182,000 

276,000 

[Form  2.] 
MORTGAGES. 


1904 

Dr. 

Cr. 

Balance 

Jan. 

Feb. 

March 

April 

May 

June 

0 
0 

Transactions  for  month 
Transactions  for  half  year 

12,000 
10,000 
50  000 
20.000 
5,000 
10,000 

00 
00 
00 
00 
00 
00 

7,000 

32000 

40.000 

12,000 

3,000 

00 

00 
00 
00 
00 

169,000 
174.000 
184,000 
202,000 
182,000 
175,000 
182,000 

00 
00 
00 
00 
00 
00 
00 

107,000 

00 

94,000 

00 

4-  13,000 

00 

July 

1 
i 

182  000 

00 

Rkal  Estate  Mortgagks.  67 

151. — In  order  to  keep  the  fullest  control  of  the  interest 
accruing  and  falling  due  periodically,  it  is  useful  to  keep  tab- 
ular registers,  classifying  the  mortgages,  first,  by  rates  of 
interest;  and  second,  by  the  months  in  which  the  interest  comes 
due.  Those  investors  who  require  all  interest  to  be  paid  at  the 
same  date  can  dispense  with  the  latter.  The  two  presentations 
or  developments  may  be  on  opposite  pages,  both  proved  by  the 
same  totals. 

Mortgages  CivAssieied  by  Rates  of  Interest. 


Date 

Total          Changes         zy^%             4% 

^%% 

5% 

6% 

1904  Jan. 

169,000                        11,000      43,000 
7,000        262  —                        7,000 

162,000 
12,000        984  + 

50,000 
12,000 

60,000 

5,000 

Feb. 

174,000                        11,000      36,000 

38,000 

60,000 

5,000 

Mortgages  Classified  by 

Dates. 

Date 

Total       Changes         J  J            FA          MS 

;        AC 

MN 

JD 

1904  Jan. 

169,000                   23,000    30,000    4,000    8,000 
7,000    262 -~ 

162,000 
12,000    984+                   12,000 

90,000 

14,000 
7,000 

Feb. 

174,000                   23,000    42,000    4,000    8,000 

90,000 

7,000 

The  numbers  in  the  column  headed  ' '  Changes  ' '  are  the 
serial  numbers  of  the  respective  Mortgages. 


68  Thb  Accountancy  of  Investment. 

CHAPTER  XI. 

Loans  on  Coi.i<aterai.. 

152. — Short- time  investments  are  often  made  upon  the 
security  or  pledge  of  bonds,  stocks,  goods  or  other  personal 
property  valued  at  more  than  the  amount  of  the  loan.  Fre- 
quently these  are  payable  on  demand,  and  are  known  as  "  call- 
loans."  It  is  evident  that  the  rate  of  interest  may  be  re- 
adjusted every  day,  or  as  often  as  either  party  is  dissatisfied,  and, 
if  an  agreement  cannot  be  reached,  the  loan  will  be  paid  off. 
Hence,  no  premium  nor  discount  will  occur  in  this  kind  of  in- 
vestment, and,  as  in  the  case  of  mortgages,  we  need  only  con- 
cern ourselves  with  principal  (at  par)  and  interest. 

153. — The  accountancy  of  loans  is  even  simpler  than  that 
of  mortgages,  and  we  need  only  give  three  models,  for  Principal 
account.  Interest  account,  and  Register  of  Collateral.  The 
latter,  at  least,  is  often  kept  on  cards  or  on  envelopes,  and  there 
is  great  danger  of  the  history  becoming  confused  and  unintellig- 
ible through  erasures  and  changes  in  the  amounts  of  collateral, 
when  substitutions  are  made.  When  part  of  a  certain  security 
is  withdrawn,  the  entire  line  should  be  ruled  out,  and  the 
reduced  quantity  re-written  on  a  new  line.  When  a  card 
becomes  at  all  complicated,  it  is  better  to  insert  a  fresh  one, 
re- writing  all  collateral,  but  keeping  the  former  card  with  it 
until  the  loan  is  entirely  liquidated. 

154. — The  Interest  account  may  be  kept  concurrent  with 
the  Principal  account  —  that  is,  using  up  the  same  number  of 
lines  in  each.  In  the  suggested  form  there  is  a  column  for  in- 
terest accrued  as  well  as  for  interest  due.  The  interest  accrued 
column  is  merely  a  preparatory  calculation  column  entered  up 
at  each  change  of  rate  or  principal,  so  that  there  may  be  only 
one  computation  to  make  when  the  interest  becomes  due.  With 
this  exception  the  mechanism  of  the  Loan  Ledger  is  the  same  as 
that  of  the  Mortgage  Ledger,  and  the  General  Ledger  account  of 
loans  will  be  similar  to  that  of  mortgages. 

155. — As  the  principal  and  the  interest  in  bond  accounts 
are  so  intimately  connected,  it  will  be  advisable  to  consider  the 
account  of  interest-revenue  more  fully  before  taking  up  the 
subject  of  bond  accounts. 


lyOANS   ON   C0I.I.ATKRAL. 


Oife 


O 


O 

O 


S 
1 

H- 1 

CO 

0 
0 

.2 

^ 

> 

1 

6 

S 

70  Thb  Accountancy  of  Investment. 

CHAPTER  XII. 

Interest  Accounts. 

156. — Interest  is  earned  and  accrues  every  day;  then,  at 
convenient  periods,  it  matures  and  becomes  collectible;  then  or 
thereafter  it  is  collected  and  takes  the  form  of  cash.  These 
three  stages  may  be  represented  by  the  book-keeping  formulas: 

1.  Interest  Accrued  /  Interest- Revenue. 

2.  Interest  Due  /  Interest  Accrued. 
S.     Cash  I  Interest  Due. 

Frequently  we  see  the  three  accounts,  Interest-Revenue,  In- 
terest Accrued  and  Interest  Due,  are  confused  under  the  one 
title  ' '  Interest, ' '  although  they  have  three  distinct  functions. 
1.  Interest- Revenue  (which  alone  may  be  termed  simply 
"  Interest ")  shows  how  much  interest  has  been  earned  during 
the  current  fiscal  period.  2.  The  balance  of  Interest  Accrued 
shows  how  much  of  those  earnings  and  of  previous  earnings 
has  not  yet  fallen  due.  3.  The  balance  of  Interest  Due  shows 
how  much  of  that  which  has  fallen  due  remains  uncollected. 

157. — The  first  of  the  three  entries  in  Article  149  is  the 
only  one  which  imports  a  modification  in  the  wealth  of  the 
proprietor;  the  other  two  are  merely  permutative,  representing 
a  shifting  from  one  kind  of  asset  to  another.  It  is  not  the  mere 
collecting  of  interest  which  increases  wealth ;  nor  is  it  merely 
the  coming-due  of  the  interest:  it  is  the  earning  of  it  from  day 
to  day. 

158. — Interest  Accrued  need  not,  and  cannot  conveniently, 
be  computed  on  each  unit  of  investment,  as  we  have  already 
stated.  But  it  can  readily  be  computed  on  all  investments  of 
the  same  kind  and  rate  of  interest,  and  the  aggregate  (say  for 
a  month)  will  be  the  amount  of  the  entry  * '  Interest  Accrued  / 
Interest- Revenue. "  Or  a  daily  rate  for  the  entire  investments 
(or  entire  class)  may  be  established,  and  this  is  used  without 
change,  day  after  day,  until  some  change  in  the  principal  or  in 
the  rate  causes  a  variation  of  the  daily  increment.  The  most 
complete  and  accurate  method  is  to  keep  a  double  register  of 
interest  earned:  Jirst,  by  daily  additions;  second,  by  monthly 
aggregates,  classified  under  rates  and  time. 


Interest  Accounts. 


71 


159. — To  exemplify  this,  we  will  take  a  period  of  ten  days 
instead  of  a  month,  and  assume  that  the  investments  are  in 
mortgages  only.  On  the  first  day  of  the  period  there  are 
$100,000  running  at  4%,  $60,000  at  4>^%,  and  $150,000  at  5%. 
On  the  second  day,  $10,000  at  4%  is  paid  off,  and  on  the  fifth 
day  $5,000  at  5%.  On  the  seventh  day  a  loan  of  $15,000  is 
made  at  4>^%,  and  one  of  $6,000  at  5%. 

160. — We  begin  by  establishing  the  daily  rate  as  follows: 

One  day  at  4%   on   $100,000 11.11,11 

One  day  at  4)4%  on  $  60,000 7 .50 

One  day  at  5%  on  $150,000 .20.83,33 

Daily  rate 39.44,44 


161.— The  decimals  are  carried  out  two  places  beyond 
cents,  and  only  rounded  in  the  total.  The  Daily  Register  will 
then  be  conducted  as  follows: 

Daii,y  Register  of  Interest  Accruing. 


Date 

No. 

Principal 
I,ess 

Principal 
More 

Rate 

1 
2 

3 

4 
5 

6 

7 

647 

453 

981 
982 

10,000 
5,000 

15,000 
6,000 

4 
5 
5 

39.44,44 
1.11,11 

38.33,33 
69,44 

37.63.88 
1.87.5 
.83,33 

39.44.44 
39.44,44 

38.33,33 
38.33,33 
38.33,33 

37.63,88 
37.63,88 

8 

9 
10 

40.34.72 
40.34,72 
40.34,72 

15.000 

21,000 

390.21 

Balances 

at  Close 

90,000 

75,000 

151,000 

4 
5 

Proof 
of  Rate 

One  day 

10. 

9.37,5 
20.97,22 

316,000 

40.34,72 

162. — The  Monthly  Register  or  Summary  takes  up,  first, 
the  mortgages  upon  which  payments  are  made,  then  those  re- 
maining to  the  end  of  the  month,  whether  old  or  new.  Its 
result  will  corroborate  that  of  the  Daily  Register. 


72  Thk  Accountancy  of  Invkstmknt. 

Monthly  Summary  of  Interest  Accruing. 


Date 

No. 

Paid  off 

Remaining 

Rate 

Days 

2 
5 

7 

10 

647 
453 
981 
982 

10  000 
5,000 

15,000 

6,000 

90,000 

60  000 

145,000 

4 

5 

^% 
5 

4  ■ 

W2 

5 

2 
5 
3 
3 
10 
10 
10 

2.22,22 
3.47,22 
5.62.5 
2.50 

100.00 
75.00 

201.38,88 

316,000 

390.21 

163. — The  Daily  and  Monthly  Registers  of  Interest  Earned 
may  be  in  separate  books  or  in  one  book — preferably  the  latter 
in  most  cases.  A  convenient  arrangement  would  be  to  use  two 
confronting  pages  for  a  month,  one  and  one-half  for  the  daily, 
and  one-half  for  the  monthly.  If  an  accurate  daily  statement 
of  affairs  is  kept,  probably  the  Daily  Interest  Accrued  will  form 
part  of  that  system.  Again,  the  interest  on  Mortgages,  on 
Bonds,  on  lyoans,  on  Discounts,  may  be  separated  or  be  all 
thrown  together.  In  all  such  respects  the  individual  circum- 
stances must  govern,  and  no  precise  forms  can  be  prescribed. 
Our  main  contention  is  that  in  some  manner  interest  should  be 
accounted  for  When  Earned  rather  than  When  Collected, 
or  W^hen  due. 

164. — The  General  I,edger  accounts  of  Interest,  Interest 
Accrued  and  Interest  Due  will  now  be  exemplified  in  simple 
form  as  to  mortgages  only.  It  is  easier  to  combine  the  several 
kinds  of  interest,  when  carrying  them  to  the  Profit  and  Loss 
account,  than  to  separate  them  if  they  are  all  thrown  in  together 
at  first. 

Interest  Revenue. 
Mortgages. 


1904 
June 

30 

Carried  to  Profit 
and  I^oss 

4270 

60 

1904 
Jan. 
Feb. 
March 
April 
May 
June 

1-31 
1-28 
1-31 
1-30 
1-31 
1-30 

Total  Earnings 
<t              it 

654 

708 
723 
756 
719 
708 

58 
25 
33 
67 
44 
33 

4270 

60 

4270 

60 

Interest  Accounts. 


73 


Interest  Accrued. 
Mortgages. 


1904 

Jan. 

0 

Balance 

2362 

50 

1-31 

Earnings 

654 

58 

Jan. 

1-31 

Due 

1272 

60 

Feb. 

1-28 

708 

25 

Feb. 

1-28 

" 

125 

00 

March 

17 

Cash  for  Accrued 
on  No.  987 

58 

33 

1-31 

Earnings 

723 

34 

March 

1-31 

875 

00 

April 

1-30 

756 

67 

April 

1-30 

•' 

625 

00 

May 

1-31 

'• 

719 

44 

May 

1-31 

" 

1200 

00 

June 

1-30 

" 

708 

33 

June 

1-30 

" 

65 

00 

0 

Balance 

30 

Balance 

2528 

94 

6691 

44 

6091 

44 

July 

2528 

94 

Interest  Due. 
Mortgages. 


1904 

Jan. 

0 

Balance 

125 

00 

1-31 

Due 

1272 

50 

Jan. 

1-31 

Collections 

1325 

00 

Feb. 

1-28 

125 

00 

Feb. 

1-28 

197 

50 

March 

1-31 

875 

00 

March 

1-31 

850 

00 

April 

130 

625 

00 

April 

1-30 

600 

00 

May 

1-31 

1200 

00 

May 

1-31 

1200 

00 

June 

1-30 

65 

00 

June 

1  30 

100 

00 

0 

Balance 

30 

Balance 

15 

00 

4287 

50 

4287 

50 

July 

15 

165. — There  is  one  entry  in  Interest  Accrued  account 
whicli  does  not  arise  from  earnings  :  the  accrued  interest  on 
Mortgage  No.  987,  which  is  paid  for  in  cash  on  March  17, 
the  mortgage  not  having  been  made  direct,  but  purchased  from 
a  previous  holder.  This  case  occurs  frequently  in  Bond  ac- 
counts, but  not  so  often  in  Mortgages. 


74 


The  Accountancy  of  Investment. 


CHAPTER  XIII. 

Bonds  and  Simii^ar  Securities. 

166. — The  investments  heretofore  considered  are  interest 
bearing,  but  bear  no  premium  nor  discount;  the  variation  from 
time  to  time  is  in  the  rate  of  interest,  while  the  principal  is 
invariable.  When  we  consider  investments  whose  price  fluc- 
tuates, while  the  cash  rate  of  interest  is  constant,  the  problem 
is  more  difficult,  because  there  are  several  prices  which  it  may- 
be desired  to  record,  viz. ,  the  original  cost,  the  market  value, 
the  par  and  the  book  value  or  amortised  value.  The  original 
cost  and  the  par  are  the  extremes:  one  at  the  beginning,  and 
one  at  the  end  of  the  investment.  The  book  values  are  inter- 
mediate to  these,  and  represent  the  investment  value,  falling  or 
rising  to  par  by  a  regular  law,  which  maintains  the  net  income 
at  the  same  rate.  The  market  value  is  not  an  investment  value, 
but  a  commercial  one ;  it  is  the  price  at  which  the  investor  could 
withdraw  his  investment,  but  until  he  has  done  so,  he  has  not 
profited  by  its  rise,  nor  lost  by  its  fall.  So  long  as  he  retains 
his  investment,  the  market  value  does  not  affect  him  nor  should 
it  enter  into  his  accounts.  It  is  valuable  information,  however, 
from  time  to  time,  if  he  has  the  privilege  of  changing  invest- 
ments, or  the  necessity  of  realizing. 

167. — The  account  of  principal,  showing  at  each  half  year 
the  result  of  amortisation,  is  very  suitably  kept  in  the  three- 
column  or  balance-column  form  recommended  in  Article  134  for 
mortgages.  Thus,  the  history  of  the  bonds  in  Schedule  G 
would  be  thus  recorded  in  ledger  form  : 

$100,000  Smithtown  5's  op  May  1,  1914. 


Date 

Dr. 

Cr. 

Balance 

1904  May  1 
Nov.  1 

1905  May  1 
Nov.  1 

1906  May  1 

Purchased  from  A.  B.  &  Co. 
Amortisation  (4%) 

104,500 

410.97 
419.19 
427.57 
436.12 

104,089.03 
103,669.84 
103.242.27 
102,806.15 

168. — In  case  of  an  additional  purchase  the  account  will, 
of  course,  be  debited  and  cash  credited.  It  will  then  be  neces- 
sary to  reconstruct  the  schedule  from  that  point  on.     This  may 


Bonds  and  Simii^ar  Securities. 


75 


be  done  in  either  of  two  ways  :  1.  Make  an  independent 
schedule  of  the  new  purchase,  and  then  consolidate  this  with 
the  old  one,  adding  the  terms.  2.  Add  together  the  values 
of  the  old  and  new  bonds  at  the  next  balance  date;  find  what 
the  basis  of  the  total  is,  eliminate  any  slight  residue  (Articles 
106,  107,  108),  and  proceed  with  the  calculation. 

169. — In  case  of  a  sale,  the  procedure  is  different.  Instead 
of  crediting  the  Bond  account  by  cash,  it  is  best  to  transfer  the 
amount  sold  to  a  Bond  Sales  account  at  its  book  value  com- 
puted down  to  the  day  of  sale ;  Bond  Sales  account  will  then 
show  a  debit,  and  the  cash  proceeds  will  be  credited  to  the  same 
account.  The  resultant  will  show  a  gain  or  loss  on  the  sale, 
and  at  the  balancing  date  the  account  will  be  closed  into  l^rofit 
and  I^oss.  Thus,  in  the  example  in  Article  165,  we  will  sup- 
pose a  sale  on  August  1,  1906,  of  half  the  $100,000  at  102.88, 
or  $51,440.  We  find  the  book  value  of  the  $50,000  on  August 
1,  which  is  $51,291 .  86;  we  transfer  this  to  the  debit  of  the  Bond 
Sales  account  in  the  General  Ledger,  which  account  we  credit 
with  the  $51,440  cash  proceeds.  Bond  Sales  is  purely  a  Profit 
and  Loss  account,  and  at  the  proper  time  will  show  the  actual 
profit  realized  on  the  sale,  $51,440  —  $51,291 .  86  =  $148 .  14. 

(Bond  Ledger.) 
$100,000  Smithtown  5's  oe  May  1,  1914. 


Date 

Dr. 

Cr. 

Balance 

1904  May  1 
Nov.  1 

1905  May  1 
Nov.  1 

1906  May  1 
Aug.  1 

Nov.  1 

Purchased  of  A.  B.  &  Co. 
Amortisation 
*« 

„-_.-  — ** — 

Sale  to  C.  D.  &  Co. 
$50,000  @  102.88 
Amortisation  on  $50,000 
'*             on  balance 

104,500 

410.97 
419.19 

4i7.67 
436.12 

51i291.86 
111.21 
222.43 

104,089.03 
103.669.84 
103.242.27 
102,806.16 

51,514.29 

51,403.08 
51,180.65 

1904 
Aug.  1 


(Generai,  Ledger.) 
Bond  Sales. 


Smithtown  5's 


51,291.86 


Aug.  1 


Proceeds 


lTr»^Ui- 


r 


-'(aV 


i^'iLc 


51,440.00 


0^fi<^^^^^' 


76  The  Accountancy  of  Inv:sstment. 

To  adjust  the  profit  on  the  Bond  account  itself  would  be  as 
unphilosophical  as  the  old-fashioned  Merchandise  account  be- 
fore the  Sales  account  was  introduced,  and  even  more  awkward. 

170. — Besides  the  book  value,  the  par  is  also  needed  because 
the  cash  interest  is  reckoned  upon  the  par.  For  some  purposes, 
also,  the  original  cost  is  useful  to  be  shown.  We  must,  there- 
fore, provide  means  for  exhibiting  these  three  values:  the  par, 
the  original  cost  and  the  book  value.  A  mere  memorandum  of 
par  and  cost  at  the  top  would  be  sufficient  where  the  group  of 
bonds  in  question  will  all  be  held  to  the  same  time;  but  this  is 
not  always  the  case,  and  provision  must  be  made  for  increase 
and  decrease.  The  three-column  form  of  ledger  spoken  of  in 
Article  158,  constantly  exhibiting  the  balance,  is  the  most 
suitable  for  this  purpose  also.  But  if  we  endeavor  to  display  all 
of  these  forms  side  by  vSide,  we  require  nine  columns,  and  this 
makes  an  unwieldy  book.  I  have,  therefore,  come  to  the  con- 
clusion that  the  most  practical  way  is  to  abandon  the  use  of 
debit  and  credit  columns,  and  proceed  by  addition  and  sub- 
traction, or  in  what  the  Italians  term  the  scalar  (ladder-like) 
form,  which  gives  a  perfectly  clear  result,  especially  if  the 
balances  are  all  written  in  red.  Headed  by  a  description  of 
the  bonds,  and  embracing,  also,  a  place  for  noting  the  market 
value  at  intervals  (not  as  matter  of  account,  but  of  information) 
the  Principal  account  will  appear  as  shown  on  page  77. 

171. — As  far  as  the  Bond  I^edger  is  concerned,  the  transfer 
of  the  $50,000  sold  to  Sales  account  is  final;  we  have,  however, 
in  the  example  indicated  a  way  of  incorporating  a  statement 
of  the  profit  or  loss  in  the  margin  for  historical  purposes.  The 
amortisation  of  Nov.  1  is  composed  of  two  parts:  3  months  on 
$50,000  sold,  $111.21,  and  the  regular  6  months  on  $50,000 
retained,  $222.43.  In  the  example  in  Article  167  these  are 
entered  separately;  either  method  may  be  pursued,  but  on  the 
whole  there  are  greater  advantages  in  postponing  all  entries  of 
amortisation  till  the  end  of  the  half  year.  The  three  months' 
amortisation  of  the  bonds  sold  is  in  effect  implied  in  the  price 
$51,291.86  reduced  from  $51,403.07,  the  half  of  $102,806.15, 
but  it  need  not  be  entered  till  Nov.  1. 

172. — The  Register  of  Interest  Due  on  Bonds  is  conducted 
on  precisely  the  same  principles  as  that  described  for  Mortgages 


Bonds  and  Similar  Securities. 


77 


be 

^     a 
O     < 


^ 

00 

^ 

^ 

1 

^ 

(U 

'■k^ 

Q 

«o 

a 

>^ 

a 

V 

CIS 

(U 

« 

Jz; 

I 


I 

CO 


^  ^ 


^    ^ 
^     (^ 


I 


>Q 


^     Ji        (U 


-  p. 


^  ^   ^   6 


,2i       «1 


CO 

t>^ 

CO 

cd 

Ph 

^ 

^ 

a 

O 

s 

W 

;z; 

S       8 

§       i 

3       § 


SS 

50-* 


Of* 


35  <* 

(N  to 


8^ 

is 


£•§ 

5£ 

M 

a 

Q 

.2 

a 

S 

o 

r- 

w< 

B 

Cfl 

< 

^^    ^ 


78 


Ths  Accountancy  of  Investment. 


in  Article  141;  in  fact,  they  are  but  sub-divisions  of  the  same 
Register.     Of  course,  the  cash  interest  is  alone  considered. 

173. — The  Interest  pages  of  the  Bond  lycdger  are  also 
similar  to  those  of  the  Mortgage  Ledger  (Article  136),  but  the 
dates  of  interest  due  may  be  printed  in  advance,  there  being 
but  little  chance  of  partial  payments  disturbing  their  orderly 
arrangement. 

174. — The  paging  of  the  Bond  Ledger  will  probably  be  geo- 
graphical, as  far  as  possible,  in  respect  to  public  issues,  and 
alphabetical  in  respect  to  those  of  private  corporations.  The 
loose-leaf  plan  permits  an  indefinite  number  of  classifications 
to  choose  from.  The  date-tags  suggested  in  Article  144  are 
especially  useful  for  pointing  out  dates  for  interest  falling 
due,  as  "J J,"  '*FA,"  etc. 

175. — The  entries  of  amortisation  are  made  directly  from 
the  schedules  of  amortisation,  the  preparation  of  which  has 
been  fully  taught  in  Chapter  VIII.  But  it  is  necessary,  also, 
to  make  up  a  list  of  these  several  amortisations  in  order  to  form 
the  General  Ledger  entry: 

Amortisation  /  Bonds, 
or^  Amortisation  /  Premiums, 
according  to  the  form  of  the  General  Ledger.  This  list  should 
be  in  the  same  order  as  the  Bond  Ledger.  Probably  the  most 
practical  way  is  to  combine  it  with  the  trial-balance  of  the  Bond 
Ledger,  thus  giving  at  each  fiscal  period  a  complete  list  of  the 
holdings,  which  may  give  the  par,  cost,  book  and  market  values, 
the  titles  of  the  securities  being  written  but  once. 


Bond  Statement  for  the  Hai,e  Year  Ending. 


Name  and 
Description 


Amorti- 
sation 


Book  Value 


Par  Value 


Original 
Cost 


Market 
Value 


The  total  of  the  second  column  will  form  the  entry  for  amorti- 
sation. The  next  three  columns  will  corroborate  the  General 
Ledger  balances. 


Bonds  and  SimiIvAr  Securities.  79 

We  have  provided  in  this  form  for  amortisation  only  and 
not  for  accumulation  on  bonds  below  par.  Where  the  latter 
values  are  few  in  number  they  may  be  embraced  in  the  same 
column,  but  distinguished  as  negatives  by  being  written  in  red 
or  encircled.  If  the  bonds  below  par  are  numerous  there  should 
be  two  columns  :  ' '  amortisation  ' '  and  *  *  accumulation. ' ' 

176. — While  the  book  value  is  the  proper  one  to  be  intro- 
duced into  the  General  lycdger,  the  par  is  very  necessary,  and 
sometimes  the  cost,  and  these  requirements  inevitably  introduce 
some  complexity.    There  are  two  modes  of  effecting  the  purpose  : 

I.  By  considering  the  par  and  cost  as  extraneous  infor- 
mation and  ruling  side  columns  for  them  beside  the 
book  value. 

II.  By  dividing  the  account  into  several  accounts,  by  the 
proper  combination  of  which  the  several  values  may  be 
obtained. 

177. — Plan  I.  will  preserve  the  conformity  of  the  Bonds 
account  with  the  Bond  I^edger  better  than  the  other.  The 
Bonds  account  may,  if  necessary,  be  extended  across  both  pages 
of  the  ledger,  to  allow  for  three  debit  and  three  credit  columns, 
if  all  are  required. 

178.— Plan  II.  will  commend  itself  more  to  those  having  a 
repugnance  to  introducing  into  the  General  Ledger  any  figures 
beyond  those  actually  forming  part  of  the  trial-balance.  The 
theory  on  which  it  is  based  is  that  the  premium  is  not  part  of 
the  bond,  but  is  a  sum  paid  in  advance  for  excess-interest,  while 
the  discount  is  a  rebate  returned  to  make  good  deficient  interest. 
This  is  a  perfectly  admissible  way  of  looking  at  the  matter, 
especially  from  the  personalistic  point  of  view;  for  the  debtor 
does  not  owe  us  the  premium  and  has  nothing  to  do  with  it. 
Still  the  other  view,  which  regards  the  investment  as  a  whole, 
is  also  correct,  and  we  may  adopt  whichever  is  most  suitable 
to  our  purposes. 

179. — If  original  cost  is  disregarded,  or  deemed  easily 
obtainable  when  required,  the  accounts  may  be 

1.  Bonds  at  Par. 

2.  Premiums. 

3.     Discounts. 
or,  1.     Bonds  at  Par. 

2.     Premiums  and  Discounts. 


8o  Ths  Accountancy  of  Investment. 

If  premiums  and  discounts  are  kept  separate,  Premiums  account 
must  always  show  a  debit  balance,  being  credited  for  amorti- 
sation ;  Discounts  account  must  show  a  credit  balance,  being 
debited  for  accumulation.  If  the  two  are  consolidated,  the  net 
amortisation  only  will  be  credited  (see  Art.  173);  or,  if  the 
greater  part  of  the  bonds  were  below  par,  the  net  accumulation 
only  would  be  debited.  The  choice  between  one  account  and 
two  for  premiums  and  discounts  would  be  largely  a  question  of 
convenience. 

The  management  of  such  a  double  or  triple  account  is 
obvious,  entries  of  transactions  being  divided  between  par  and 
premiums,  or  par  and  discounts,  but  we  give  on  pages  83  and 
following,  an  example  of  each. 

We  shall  hereafter  confine  the  discussion  to  premiums, 
leaving  the  cases  of  discount  to  be  determined  by  analogy. 

180. — Where  it  is  deemed  necessary  to  keep  account  of  cost 
also,  as  well  as  of  par  and  book  value,  the  diflSculty  is  some- 
what greater,  as  we  have  a  valueless  or  extinct  quantity  to 
record,  namely  so  much  of  the  original  premium  on  bonds  still 
held  as  has  not  yet  been  absorbed  in  the  process  of  amortisation. 
This  carrying  of  a  dead  value,  which  is  somewhat  artificial, 
necessitates  the  carrying,  also,  of  an  artificial  annulling  or  off- 
setting account,  the  sole  function  of  which  is  to  express  this 
departed  value.  We  may  call  this  credit  account  '  'Amortisation 
Fund."  It  is  analogous  to  Depreciation  and  Reserve  Funds. 
The  part  of  the  premiums  which  has  been  extinguished  by  the 
Amortisation  Fund  may  be  designated  as  '  *  Premiums  Amor- 
tised," or  "Ineffective  Premiums,"  while  the  live  premiums 
may  be  styled  "  Effective  Premiums,"  being  what  in  Art.  177 
we  called  simply  "  Premiums."  A  double  operation  takes  place 
in  these  accounts  :  first,  the  absorption  of  effective  premiums  by 
lapse  of  time  ;  and  second,  the  rejection  of  ineffective  premiums 
upon  redemption  or  sale. 

181. — There  are  two  ways  of  carrying  on  these  accounts, 
differing  as  to  Premiums.  We  may  keep  two  accounts :  '  'Effec- 
tive Premiums"  and  "Amortised  Premiums,"  or  we  may 
combine  these  in  one,  * '  Premiums  at  Cost. ' '  The  entire  scheme 
will  be  :  a.     Bonds  at  Par. 

b.     Premiums  at  Cost. 

e.     Amortisation  Fund. 


Bonds  and  Simii^ar  Securities.  8i 

or^  a.     Bonds  at  Par. 

c.  Effective  Premiums. 

d.  Amortised  Premiums. 

e.     Amortisation  Fund. 

**a*'  will  in  both  schemes  be  the  same;  "e"  will  also  be 
the  same,  "b"  is  the  sum,  c+d.  In  the  former,  the  cost  is 
a-f-b,  while  the  book  value  is  a+b — e.  In  the  latter  the  book 
value  is  a+c,  while  the  cost  is  a+c+d.  The  former  gives  the 
cost  more  readily  than  the  latter,  and  the  book  value  less 
readily.  The  former  might  be  considered  the  more  suitable  for 
a  trustee  ;   the  latter,  for  an  investor. 

182. — Account  a.  Bonds  at  Par,  is  debited  for  par  value  of 
purchases  and  credited  for  par  value  of  sales.  Its  two  only 
entries  are : 

Bonds  at  Par/Cash. 

Cash/Bonds  at  Par. 

183. — In  case  of  purchase  at  a  premium,  the  premium  is 
charged  to  Premiums  at  Cost  or  to  Effective  Premiums,  as  the 
case  may  be,  there  being  no  ineffective  premium  at  this  time. 

184. — When  premiums  are  written  off,  on  the  first  plan, 
there  is  but  one  entry  :  crediting  the  Amortisation  Fund  and 
debiting  the  Profit  and  I^oss  account  or  its  sub-division. 

Amortisation/ Amortisation   Fund. 

185. — The  second  plan  involves  not  only  this  process,  but 
a  transfer  from  Effective  to  Amortised  Premiums.  Thus  the 
aggregate  of  premiums  written  off  is  posted  four  times  as  a 
consequence  of  the  separation  of  premiums  at  cost  into  two 
accounts  : 

Premiums  Amortised/ Effective  Premiums. 
Amortisation/Amortisation  Fund. 

186. — The  word  "Amortisation"  has  been  used  in  the 
specimen  entries  as  the  title  of  an  account  tributary  to  Profit 
and  I,oss.  At  the  balancing  period  it  may  be  disposed  of  in 
either  of  two  ways'. :  it  may  be  closed  into  Profit  and  lyoss  direct ; 
or  it  may  be  closed  into  Interest  account,  the  balance  of  which 
will  enter  into  Profit  and  Loss  at  so  much  lessened  a  figure. 
In  the  former  method  the  Profit  and  I^oss  account  will  show,  on 


82  Thk  Accountancy  of  Investment. 

the  credit  side,  tlie  gross  cash-interest,  and  on  the  debit  the 
amount  devoted  to  amortisation  ;  the  second  method  exhibits 
the  net  income  only.  Whether  it  be  preferable  to  show  both 
elements,  or  only  the  net  resultant,  will  be  determined  by 
expediency. 

187. — In  Articles  169  and  171  we  discussed  two  methods  of 
keeping  account  of  amortisation  :  the  first  (169),  where  any  in- 
cidental amortisation  occurring  in  the  midst  of  the  period  is 
at  once  entered;  the  second,  exemplified  in  171,  where  all  such 
entries  are  deferred  to  the  end  of  the  period,  and  comprised  in 
one  entry  in  the  General  I^edger. 

If  the  latter  method  be  adopted,  the  Amortisation  account 
may  be  dispensed  with  altogether,  and  the  total  amount 
amortised  (which  is  credited  to  Bonds,  or  to  Premiums,  or  to 
Amortisation  Fund)  may  be  debited  at  once  to  Profit  and  I,oss 
or  to  Interest,  without  resting  in  a  special  account.  A  single 
item,  of  course,  needs  no  machinery  for  grouping. 

188.— Irredeemable  Bonds  (Art.  126)  merely  lack  the 
element  of  amortisation,  and  require  no  special  arrangement  of 
accounts.  The  par  is  purely  ideal,  as  it  never  can  be  demand- 
ed and  is  merely  a  basis  for  expressing  the  interest  paid. 
What  the  investor  buys  is  a  perpetual  annuity.  If  he  buys 
such  annuity  of  $6  per  annum,  it  is  unimportant  whether  it  is 
called  6%  on  $100  principal,  or  4%  on  $150  principal ;  and  this 
$150  may  be  the  par  value,  or  it  may  be  $100  par  at  50% 
premium,  or  $200  par  at  25%  discount.  The  par  value  is 
really  non-existent ;  and  this  illustrates  the  absurdity  of  re- 
ducing even  redeemable  securities  to  par,  which  is  practised  by 
some  investors,  par  being,  except  at  the  moment  of  maturity,  an 
unreal  sum. 

189. — We  will  now  give  examples  of  the  two  plans  for  the 
General  Ledger  outlined  in  Articles  176  to  187.  We  will  sup- 
pose that  on  Jan.  1,  1901,  the  following  lots  of  bonds  are  held: 


Bonds  and  Similar  Securitibs. 


83 


Par 
100,000 


100.000 


10.000 


210,000 


January  1,  1901. 

Book  Vai^ue 
5%  Bonds,  J.  J., 

due  Jan.  1,  1911,  net  2.7%;  value 120,039.00 

original  cost,  124,263.25 

3%  Bonds,  M.  N., 

due  May  1,  1904,  net  4%;  value 96,909.10 

original  cost,  93,644.28 

4%  Bonds,  A.  O., 

due  Oct.  1,  1902,  net  3%;  value 10,169.19 

original  cost,  10,250.00  

Totals  227,117.29 


The  premiums  on  the  5%  and  4%  bonds  amount  to 
$20,208.19.  The  discount  on  the  3%s  is  $3,090.90.  The  net 
premium  is  $17,117.29.    The  total  original  cost  was  $228,157.53, 


Dr. 


P1.AN  I  (Art.  176)  For  Genkrai.  I^bdgkr. 
Bonds  Account. 


Cr. 


1901 

Par 

Cost 

1901 

June  30,  A  mortisation 
Dpc.  31, 

1902 
June  30,            " 
Oct.     1,  Redeemed 
Dec.  31,  Amortisation 

1903 
June  30, 
Dec.  31, 

"        Balances 

Par 

Cost 

Jan.  0,  Balances 

210,000.00 

228,157.53 

227,117  29 

10,000.00 
200,000.00 

10,250.00 
217,-907.53 

488.76 
492.59 

496  40 

10,000  00 

475  21 

453.63 

456.68 

214.254.02 

1904 

210,000.00 

228,157.53 

227.117.29 

210,000.00 

228,157.53 

227,117.29 

Jan.  0,  Balances 

200,000.01; 

217,907.53 

214,254.02 

84 


Thk  Accountancy  of  Investment. 


PI.AN  II,  1  (Art.  179),  For  General  IvEdger. 


Dr. 


Bonds  at  Par. 


Cr. 


1901 
Jan.  0,  Balance. 


.210.000.00 


1902 
Oct.  1,  Redeemed 10,000.00 


Dr. 


Premiums. 


Cr. 


1901 
Jan.  0,  Balance. 


.20,208.19 


1901 

June  30,  Amortisation 926.94 

Dec.  31,  •'  939.54 

1902 

June  30,  "  952.28 

Dec.  31,  ♦*  940.21 

1903 

June  30,  "  927.93 

Dec.  31,  **  940.47 


Dr. 

Discounts. 

Cr, 

1901 

1901 

June  30, 

Accumulation.. . 

. . .438.18 

Jan.  0,  Balance 

3,090.90 

Dec.  31, 

({ 

....446.95 

1902 

June  30, 

(( 

....455.88 

Dec.  31, 

(( 

....465.00 

1903 

June  30, 

<( 

....474.30 

Dec.  31, 

<« 

....483.79 

Bonds  and  Simii^ar  Skcuriti^s. 


85 


P1.AN  II,  2  (Art.  179),  For  Gknkrai.  I^kdoer. 

ORIGINAI,  COST  OMITTED. 

Dr.  Bonds  at  Par.  Cr. 


1901 
Jan.  0,  Balance. 


.210,000.00 


1902 
Oct.  1,  Redeemed. 


.10,000.00 


Dr. 


Premiums  and  Discounts. 


Cr. 


1901 
Jan.  0,  Balance. 


,17,117.29 


1901 

June  30, 

Amortisation. . 

.  .488.76 

Dec.  31, 

t( 

.  .492.69 

1902 

June  30, 

t( 

...496.40 

Dec.  31, 

<c 

.  .475.21 

1903 

June  30, 

(( 

.  .453.63 

Dec.  31, 

<< 

. .  .456.68 

86 


Thb  Accountancy  of  Investment. 


PI.AN  II,  3  (Art.  184),  For  Generai.  I^edger. 

"BONDS  AT  PAR"  AS  IN  FOREGOING  PLANS. 


Dr. 


Premiums  at  Cost. 


Cr. 


1901 
Jan.  0,  Balance 18,157.53 


1903 


18,157.53 


Jan.  0,  Balance 17,907.53 


1902 
Oct.  1,  Canceled  at  Re- 
demption        250.00 

1903 

Dec.  31,  Balance 17,907.53 

18,157.53 


Dr. 


Amortisation  Fund. 


Cr. 


1902 
Oct.  1,  Canceled  at  Re- 
demption     250.00 

1903 
Dec.  31,  Balance 3,653.51 


3,903.51 


1901 

Jan.    0,  Balance 1,040.24 

June  30,  Amortisation .. .    488.76 


492.59 

496.40 
475.21 


Dec.  31, 

1902 
June  30, 
Dec.  31, 

1903 
June  30, 
Dec.  31, 

1904                              =^ 
Jan.  0,  Balance 3,653.51 


453.63 
456.68 


3,903,51 


Bonds  and  Similar  Skcuritiks. 
Plan  II,  4  (Art.  185),  For  General  I^edger. 

BY  THE  balance  COLUMN  METHOD. 


Bonds  at  Par. 


Br. 


Cr.        Balance  Dr. 


1901  Jan.  0 

1902  Oct.  1 


Balance 

Redemption. 


210,000.00 


10,000.00 


200,000.00 


Effectivb  Premiums. 


Dr. 


Cr.        Balance  Dr. 


1901  Jan.  0 
June  30 
Dec.  31 

1902  June  30 
Dec.  31 

1903  June  30 
Dec.  31 


Balance . . . 
Amortised 


17,117.29 

488.76 
492.59 

16,628.53 
16,135.94 

496.40 
475.21 

15,639.54 
15,164.33 

463.63 
456.68 

14,71070 
14,254.02 

INEEFKCTIVB  OR  AMORTISED  PREMIUMS     Dr. 


Cr.        Balance  Dr. 


1901  Jan.  0 
June  30 
Dec.  31 

1902  June  30 
Oct.  1 
Dec.  31 

1903  June  30 
Dec.  31 


Balance . . . 
Amortised . 


Canceled  by  Redemption 
Amortised 


1,040.24 

488.76 
492.59 

496.40 

475.21 

453.63 
456.68 


250.00 


1,529.00 
2,021.59 

2,517.99 
2,267.99 
2,743.20 

3,196.83 
3,653.51 


Amortisation  Fund. 

Dr. 

Cr,        Balance  Cr. 

1901  Jan.    0 
June  30 
Dec.  31 

1902  June  30 
Oct.     1 
Dec.  31 

Balance 

250 

1,040.24 

488.76 
492.59 

496.40 

475.21 

453.63 
456.68 

Amortised 

(< 

Canceled  by  Redemption. 
Amortised 

1,529.00 
2,021.59 

2,517.99 
2,267.99 
2,743.20 

3,196.83 
3,653.51 

1902  June  30 

<< 

Dec.  31 

<c 

88  The  Accountancy  of  Investment. 


CHAPTER  XIV. 

Discounted  Vai^ues. 

190. — The  securities  heretofore  considered  have  all  carried 
a  stipulated  rate  of  interest  or  annuity.  There  is  another  class 
in  which  no  periodical  interest  attaches,  but  the  obligation  is 
simply  to  pay  a  single  definite  sum  on  a  certain  date.  The 
present  value  of  that  sum  at  the  current  or  contractual  rate  of 
income  is,  of  course,  obtained  by  discounting  according  to  the 
principles  explained  in  Chapter  II.  If  the  maturity  were  more 
than  one  year  distant  at  the  time  of  discount,  it  would  be 
necessary  to  compute  the  compound  discount ;  but  in  practice 
this  never  occurs,  such  discounts  being  for  a  few  months. 

191. — The  obligations  so  treated  are  almost  invariably 
promissory  notes.  Formerly  they  consisted  largely  of  bills  of 
exchange,  hence  the  survival  in  bookkeeping  of  the  words 
"  Bills  Receivable,"  "  Bills  Payable"  and  ''  Bills  Discounted." 

192. — These  obligations  belong  rather  to  mercantile  and 
banking  accountancy  than  to  that  of  investment.  The  arrange- 
ment of  accounts  for  recording  their  amounts,  classification  and 
maturity  has  been  so  fully  treated  in  works  on  those  branches 
that  we  only  refer  to  them  here  for  the  purpose  of  illustrating 
another  phase  of  the  process  of  securing  income. 

193. — The  difference  between  the  rate  of  interest  and  the 
rate  of  discount  has  been  pointed  out  in  Chapter  II.  It  was 
there  shown  (Art.  23)  that  in  a  single  period  the  rate  of  interest 
3%  corresponds  to  a  rate  of  discount  .02913.     Hence,  if  we 


Discounted  Vai^ues.  89 

discount  a  note  at  .02913,  we  acquire  interest  at  the  rate  of  .03 
on  the  .97087  actually  invested.  The  rate  of  interest  is  always 
greater  than  the  rate  of  discount. 

It  is  usual  to  name  a  rate  of  discount  rather  than  a  rate  of 
interest  in  stipulating  for  the  acquisition  of  notes.  For  example, 
a  three  months  note  for  $1000  is  taken  for  discount  at  6% 
[per  annum].  This  means  that  .015  is  to  be  retained  by  the 
payee  from  each  dollar  and  the  amount  actually  paid  over  is 
$985.  The  income  from  this  is  the  $15,  and  by  dividing  $15  by 
$985  we  readily  ascertain  that  the  rate  of  interest  realized  is 
.0609.  It  is  sometimes  believed  that  there  is  a  kind  of  decep- 
tion in  this  ;  that  the  borrower  agrees  to  pay  6%  and  actually 
has  to  pay  6.09%.  But  this  is  not  so  :  the  bargain  is  not  to 
pay  6%  interest,  but  to  allow  6%  discount,  which  is  a  different 
thing. 

193. — Curiously,  the  lawfulness  or  unlawfulness  of  a  trans- 
action sometimes  depends  upon  the  mere  form  of  words  in 
which  it  is  expressed.  Thus,  if  I  lend  $985  to  another,  who 
promises  to  repay  $1000  at  3  months,  if  his  promise  reads 

* '  I  promise  to  pay  $1000, ' ' 
I  am  a  law-abiding  citizen  ;  but  if  he  writes 
'*I  promise  to  pay  $985  and  interest  at  6.09%  per  annum,'* 
the  statute  is  violated.     But  it  is  too  much  to  expect  regard 
for  logic  in  a  usury  law. 

194. — Notes  discounted  are  usually  entered  among  the 
assets  at  the  full  face  and  the  discount  credited  to  an  opposite 
account,  ' '  Discounts, ' '  the  latter  having  precisely  the  same  effect 
as  the  Discounts  account  used  in  connection  with  bonds.  The 
difference  of  the  two  is  the  cost. 

195. — Strictly  speaking,  the  discount  is  at  first  an  offset 
to  the  note,  and  represents  at  that  time  nothing  earned  what- 
ever ;  as  time  goes  on,  the  earning  is  effected  by  diminution  of 
this  offset,  which  is  equivalent  to  a  rise  of  the  net  value  of  the 
note,  from  cost  to  par.  We  may  represent  the  process  by 
exhibiting  the  state  of  the  accounts  at  the  initial  date  and  at 
the  end  of  each  month  up  to  maturity,  for  a  3  months'  note  for 
$1000  discounted  at  6%. 


90 


Thk  Accountancy  of  Investment. 


Note. 


1.    When  Discounted. 
Discount. 


1000.00 


Note. 


15.00 

2.    After  One  Month. 

Discount.  Interest   Revenue. 


5.00 


3.    After  Two  Months. 

Discount.  Interest  Revenue. 


1000.00 


5.00 
5.00 


15.00 


Note. 


4.     At  Maturity. 
Discount. 


5.00 
5.00 


Interest  Revenue. 


1000.00 


5.00 
5.00 
5.00 

15.00 

15.00 

15.00 

5.00 
5.00 
5.00 


195.  Notes  being  for  short  periods,  this  gradual  crediting 
of  earnings  is  usually  ignored,  and  the  Discounts  account  stands 
unaltered  until  the  balancing  period  and  is  then  closed  into 
Profit  and  lyoss.  As  most  of  the  notes  have  matured  during 
the  period,  the  result  is  correct  so  far  as  concerns  those  notes  ; 
but  as  to  the  notes  still  running  there  is  an  error,  for  the 
discount  has  not  all  been  earned.  It  is  proper  to  make  an 
adjustment  to  correct  this  error,  which  can  be  done  as  follows  : 

19^.^— Compute  interest  from  the  balancing  date  to  the 
maturity  of  each  note  on  the  par  at  the  rate  of  discount ;  sub- 
tract the  total  from  the  total  of  Discounts  account ;  transfer 
only  the  remainder  to  Interest  or  directly  to  Profit  and  Loss. 
Then  balance  the  Discounts  account,  and  the  balance  brought 
down  will  be  the  discounts  as  yet  unearned.  The  investment 
value  of  the  notes  on  hand  will  be  the  difference  between  the 
par  and  the  unearned  discount. 


The  Accountancy  of  Investment. 


91 


APPENDIX  I. 

I^OGARITHMS,   TO  12   P1.ACES,    OF  VARIOUS  RATIOS. 


RATIO 

1.005 

1.00525 

1.0055 

1.00575 

1.006 

1.00625 

1.0065 

1.00675 


I^OGARITHM 


007 

00725 

0075 

00775 

008 

00825 

0085 

00875 

009 

00925 

0095 

00975 

l.Oi 

1.0105 

1.011 

1.01125 


0115 

012 

0125 

013 

0135 

01375 

014 

1.0145 

1.015 

1.0155 

1.016 

1.01625 

1.017 

1  0175 

1.018 

1.0185 

1.01875 

1.019 

1.02 

1.0205 

1.021 

1.02125 

1.0215 


.002  166 
.002  274 
.002  382 
.002  490 
.002  597 
.002  705 
.002  813 
.002  921 
.003  029 
.003  137 
.003  245 
.003  352 
.003  460 
.003  568 
.003  675 
.003  783 
.003  891 
.003  998 
.004  106 
.004  213 
.004  321 
.004  536 
.004  751 
.004  858 
004  965 
,005  180 
.005  395 
.005  609 
.005  823 
.005  930 
.006  037 
.006  252 
.006  466 
.006  679 
.006  893 
.007  000 
.007  320 
.007  534 
.007  747 
.007  96] 
.008  067 
.008  174 
.008  600 
.008  813 
.009  025 
.009  132 
.009  238 


061  757 
081  775 
074  933 

041  244 
980  720 
893  376 
779  225 
638  280 
470  554 
276  061 
054  813 
806  825 
532  110 
230  680 
902  549 
547  730 
166  237 
758  083 
323  280 
861  842 
373  783 
317  852 
155  591 
534  621 
887  107 
512  504 
031  887 
445  360 
753  029 
867  219 
954  997 
051  369 

042  249 
927  741 
707  948 
558  602 
952  923 
417  897 
778  001 
033  336 
621  748 
184  006 
171  762 
009  052 
742  087 
069  540 
370  968 


RATIO 

1.022 

1.0225 

1.023 


lyOGARITHM 


0235 

02375 

024 

0245 

025 

0255 

,026 

02625 

1.0265 

1.027 

1.0275 

1.028 

1.02875 

1.029 

1.0295 

1.03 

1.031 

1.03125 

1.032 

1.0325 

1.033 

1.03375 

1.034 

1.035 

1.036 


03625 

037 

0375 

038 

03875 

039 

04 

0425 

045 

0475 

05 

,0525 
1.055 
1.0575 
1.06 
1.0625 
1.065 
1-0675 
1-07 


.009  450 
.009  663 
.009  875 
.010  087 
.010  193 
.010  299 
.010  511 
.010  723 
.010  935 
.011  147 
.011  253 
.011  358 
.011  570 
.011  781 
.011  993 
.012  309 
.012  415 
.012  626 
.012  837 
.013  258 
.013  363 
.013  679 
.013  890 
.014  100 
.014  415 
.014  520 
.014  940 
.015  359 
.015  464 
.015  778 
.015  988 
.016  197 
.016  511 
.016  615 
.017  033 
.018  076 
.019  116 
.020  154 
.021  189 
.022  222 
.023  252 
.024  280 
.025  305 
.026  328 
.027  349 
.02»  367 
.029  383 


895  799 
316  679 
633  712 
846  999 
914  768 
956  640 
962  737 
865  392 

664  704 
360  776 
170  127 
953  707 
443  597 
830  548 
114  659 
848  220 
374  762 
350  954 
224  705 

665  284 
961  558 
697  291 
060  328 
321  520 
522  561 
538  758 
349  793 

755  409 
543  558 

756  389 
105  384 
353  512 
036  792 
547  557 
339  299 
063  646 
290  447 
031  638 
299  070 
104  508 
459  634 
376  047 
865  265 
938  722 
607  775 
883  697 
777  685 


APPENDIX   II. 

Multipliers  for  Converting  Semi- Annual  Premiums  or  Discounts 
TO  Equivalents  for  Quarterly  Bonds.    See  Note,  P.  54. 


Net 
Income 

3%  Bond 

J>^%  Bond 

4%  Bond 

4>^%  Bond 

5%  Bond 

6%  Bond 

7%  Bond 

2.50 

.0186918 

.0109035 

.0083075 

.0070094 

.0062306 

.0053405 

.0048460 

2.55 

.0211827 

.0117062 

.0087653 

.0073325 

.0064845 

.0055259 

.0049982 

2.60 

.0242963 

.0125981 

.0092557 

.0076725 

.0067490 

.0057168 

.0051538 

2.65 

.0282994 

.0135948 

.0097825 

.0080309 

.0070247 

.0059133 

.0053129 

2.70 

.0336369 

.0147161 

.0103498 

.0084092 

.0073124 

.0061158 

.0054758 

2.75 

.0411092 

.0159869 

.0109604 

.0088091 

.0076128 

.0063245 

.0056424 

2.80 

.0523175 

.0174392 

.0116261 

.0092325 

.0079269 

.0065397 

.0058131 

2.85 

.0709980 

.0191148 

.0123475 

.0096815 

.0082556 

.0067617 

.0059867 

2.90 

.1083586 

.0210697 

.0131344 

.0101586 

.0085999 

.0069909 

.0061668 

2.95 

.2204401 

.0233800 

.0139962 

.0106665 

.0089610 

.0072275 

.0063501 

3.00 

.0261523 

.0149442 

.0112081 

.0093401 

.0074721 

.0065381 

3.05 

: 2278844 

.0295406 

.0159919 

.0117871 

.0097387 

.0077249 

.0067308 

3.10 

.1158030 

.0337759 

.0171560 

.0124075 

.0101582 

.0079864 

.0069284 

3.15 

.0784424 

.0392212 

.0184570 

.0130737 

.0106003 

.0082571 

.0071311 

3.20 

.0597619 

.0464815 

.0199206 

.0137912 

.0110670 

.0085374 

.0073392 

3.25 

.0485535 

.0566458 

.0215794 

.0145661 

.0115604 

.0088279 

.0075528 

3.30 

.0410812 

.0718922 

.0234750 

.0154055 

.0120827 

.0091292 

.0077721 

3.35 

.0357438 

.0973026 

.0256622 

.0163178 

.0126367 

.0094418 

.0079975 

3.40 

.0317407 

.1481232 

.0282139 

.0173131 

.0132253 

.0097664 

.0082291 

3.45 

.0286271 

.3005843 

.0312196 

.0184031 

.0138518 

.0101037 

.0084672 

3.50 

.0261361 

.0348482 

.0196021 

.0145201 

.0104545 

.0087121 

3.55 

.0240981 

! 3092587 

.0392710 

.0209273 

.0152344 

.0108195 

.0089640 

3.60 

.0223997 

.1567976 

.0447993 

.0223997 

.0159998 

.0111998 

.0092234 

3.65 

.0209625 

.1059770 

.0519071 

.0240452 

.0168217 

.0115963 

.0094905 

3.70 

.0197306 

.0805666 

.0613841 

.0258964 

.0177069 

.0120097 

.0097656 

3.75 

.0186629 

.0653202 

.0746517 

.0279944 

.0186629 

.0124419 

.0100493 

3.80 

.0177287 

.0551559 

.0945530 

.0303920 

.0196985 

.0128936 

.0103417 

3.85 

.0169043 

.0478956 

.1277216 

.0331662 

.0208242 

.0133662 

.0106435 

3.90 

.0161715 

.0424503 

.1940585 

.0363860 

.0220521 

.0138613 

.0109549 

3.95 

.0155159 

.0382150 

.3930687 

.0402002 

.0233969 

.0143805 

.0112766 

4.00 

.0149260 

.0348273 

.0447780 

.0248767 

.0149260 

.0116091 

4.05 

.0143920 

.0320549 

! 4029757 

.0503720 

.0265116 

.0154991 

.0119525 

4.10 

.0139065 

.0297444 

.2039616 

.0573642 

.0283280 

.0161022 

.0123080 

4.15 

.0134631 

.0277893 

.1376231 

.0663540 

.0303580 

.0167379 

.0126758 

4.20 

.0130567 

.0261134 

.1044536 

.0783402 

.0326418 

.0174089 

.0130567 

4.25 

.0126830 

.0246613 

.0845532 

.0951223 

.0352305 

.0181185 

.0134516 

4.30 

.0123379 

.0233906 

.0712855 

.1202944 

.0381887 

.0188696 

.0138611 

4.35 

.0120183 

.0222693 

.0618086 

.1622475 

.0416019 

.0196664 

.0142859 

4.40 

.0117216 

.0212726 

.0547008 

.2461535 

.0455840 

.0205128 

.0147271 

4.45 

.0114453 

.0203807 

.0491724 

.4978708 

.0502900 

.0214138 

.0151856 

4.50 

.0111874 

.0195780 

.0447497 

.0559371 

.0223748 

.0156624 

4.55 

.0109462 

.0188517 

.0411310 

; 5089964 

.0628391 

.0234021 

.0161586 

4.60 

.0107200 

.0181914 

.0381154 

.2572791 

.0714664 

.0245028 

.0166755 

4.65 

.0105075 

.0175886 

.0355637 

.1733731 

.0825586 

.0256849 

.0172143 

4.70 

.0103075 

.0170359 

.0333765 

.1314200 

.0973481 

.0269579 

.0177766 

4.75 

.0101188 

.0165274 

.0314808 

.1062679 

.1180532 

.0283328 

.0183638 

4.80 

.0099407 

.0160581 

.0298221 

.0894664 

.1491106 

.0298221 

.0189777 

4.85 

.0097722 

.0156257 

.0283585 

.0774795 

.2008728 

.0314410 

.0196201 

4.90 

.0096125 

.0152198 

.0270575 

.0684893 

.3043969 

.0332069 

.0202931 

4.95 

.0094610 

.0148441 

.0258934 

.0614968 

.6149681 

.0351410 

.0209989 

5.00 

.0093171 

.0144933 

.0248457 

.0559028 

.0372685 

.0217400 

The  Accountancy  of  Investment.  93 


APPENDIX  III. 
Summary  of  Compound  Interest  Processes. 


To  find  the  Ratio  of  Increase 
Add  1  to  the  Rate  of  Interest. 

To  find  the  Amount  ^$1 

Multiply  1  by  the  Ratio  as  many  times  as  there  are  periods. 

To  find  the  Present  Worth  of%\,or  to  discount  $1 

Divide  1  by  the  Ratio  as  many  times  as  there  are  periods. 

To  find  the  Total  Interest 

Subtract  1  from  the  Amount. 

To  find  the  Total  Discount 

Subtract  the  Present  Worth  from  1. 

To  find  the  Amount  of  an  Annuity  ^$1 

Divide  the  Total  Interest  by  the  Rate  of  Interest. 

To  find  the  Present  Worth  of  an  Annuity  of%\ 

Divide  the  Total  Discount  by  the  Rate  of  Interest. 

To  find  the  Rent  of  an  Annuity  worth  $1,  or  what  Annuity  can 
be  bought  for  $1 

Divide  1  by  the  Present  Worth  of  the  Annuity. 

To  find  what  Annuity  (Sinking  Fund)  will  produce  $1 
Divide  1  by  the  Amount  of  the  Annuity. 

To  find  the  Premium  or  Discount  on  a  Bond 

Consider  the  Difference  of  Interest  as  an  Annuity  to  be 
valued,  and  find  its  Present  Worth. 


BY  THE  SAME  AUTHOR : 

EXTENDED  BOND  TABLES 

GIVING  ACCURATE  VAI^UBS  TO  EIGHT 
PI^ACES  OF  DECIMAI.S  OR  TO  THE 
NEAREST    CENT    ON    $1,000,000      :      :      : 

New  York,  1905.  BUSINESS  PUBLISHING  CO. 


^■^■'  >^^^:^i^ii^^-^W^\>^^ 


'  ■' '^h'i?\v:'i'*ih^'''/U  :';ir;> i^-s''"?''' v!- A^P^ 


LIBRARY  USE 

RCTUKN  TO  DESK  FROM  WHICH  BORROWED 

MAIN  LIBRARY 
CIRCULATION  DEPARTMENT 

THIS  BOOK  IS  DUE^BEFORE  CLOSING  TIME 
*        ON  LAST  DATE  STAMPED  BELOW 


tKmrw-j^^^ 


im- 


ffiCQlR.    fEB2576 


LD  21-100wi-8,'34 


YC  6'^37l 


